Acoustic impedance and reflection coefficients

Part 1 — Foundations of Seismic

Learning objectives

State the definition of acoustic impedance and compute it from log data
Derive the normal-incidence reflection coefficient from impedance contrast
Interpret the sign of a reflection coefficient as polarity
Estimate relative reflector brightness by comparing impedance contrasts

Section 0.3 introduced acoustic impedance as "the thing that matters at a boundary." Now we make that quantitative. Understanding reflection coefficients is the difference between seeing patterns on seismic and understanding why those patterns are there.

Acoustic impedance, defined precisely

For a plane wave traveling through a material, the acoustic impedance Z is the ratio of pressure to particle velocity. Intuitively, it tells you how much the medium "resists" being compressed. A material that is both dense and fast (thick and stiff) has high Z. A material that is light and slow (loose, compressible) has low Z.

For seismic work, the working definition is:

$Z = \rho \cdot V_p$

where $\rho$ is bulk density (kg/m³ or g/cm³) and $V_p$ is P-wave velocity (m/s). Units of Z are kg/(m²·s) in SI or (g/cm³)·(m/s) in the mixed units we see on log displays. The numerical value matters less than the contrast across boundaries.

Real sedimentary rocks span roughly one order of magnitude in acoustic impedance:

Unconsolidated shallow sediments: $Z \approx 3000$ –5000 (g/cm³·m/s)
Average siliciclastic section (sands and shales, moderate burial): $Z \approx 6000$ –9000
Well-cemented sandstones, most carbonates: $Z \approx 9000$ –13000
Tight carbonates, salt, anhydrite, hard volcanic rock: $Z \approx 13000$ –20000

Gas-filled rocks can fall well below these ranges; evaporites (salt, anhydrite) sit at the high end. The point of listing these is to calibrate your intuition: when you compute an impedance of 2800, something unusual is happening (possibly a gas sand); when you compute 16000, you are looking at a very hard rock.

The normal-incidence reflection coefficient

Consider a seismic wave traveling straight down (normal incidence, incidence angle = 0) through medium 1 and hitting the boundary with medium 2. The fraction of amplitude that reflects back is:

$R = \dfrac{Z_2 - Z_1}{Z_2 + Z_1}$

R is a pure ratio, dimensionless, between –1 and +1. The sign matters:

R > 0: impedance increases going into the reflector. The reflected pulse has the same polarity as the incident pulse. On standard SEG-positive display this appears as a peak.
R < 0: impedance decreases going into the reflector. The reflected pulse is flipped in polarity. On standard SEG-positive display this appears as a trough.
R = 0: no impedance contrast, no reflection. The boundary is invisible to seismic.

Where does this formula come from? Pressure and particle velocity must be continuous across the boundary (mechanical continuity). Writing down those two continuity conditions and solving for the reflected-wave amplitude relative to the incident-wave amplitude gives exactly the formula above. The full derivation uses the Zoeppritz equations, which also handle non-normal incidence, mode conversions between P and S waves, and so on. Part 5 covers the full Zoeppritz machinery. For now, normal-incidence is enough — and it is what stacked seismic approximates.

What is a "bright" reflector?

A rough field calibration for reflection coefficient magnitude:

|R| < 0.05 — weak, may be lost in noise
|R| = 0.05 to 0.15 — typical sedimentary reflector
|R| = 0.15 to 0.25 — strong, clearly visible, a key interpretation marker
|R| > 0.25 — very bright. Examples: top of salt under shale, shale on tight carbonate, top of a gas sand

A reflection coefficient of 0.30 means 30% of the incident amplitude reflects. Amplitude of 0.5 would be an extraordinary value (encountered only at the sea floor or at gas-charged sands in exceptional cases). Knowing this scale prevents you from being over-impressed or under-impressed by the relative brightness of reflectors on your section.

One subtle but important point. The amplitude you read off a seismic trace is proportional to R, but also scaled by the wavelet shape, by processing gain choices, and by acquisition geometry. You rarely get to measure R directly. What you can do is compare relative brightness between reflectors in the same dataset — a reflector twice as bright as its neighbour plausibly has twice the impedance contrast, other things being roughly equal. That comparative reasoning is the bread-and-butter of qualitative interpretation.

Transmission and energy conservation

If a fraction R of amplitude reflects, the rest transmits into the lower layer. The transmission coefficient (for amplitude) is $T = 1 - R$ . Note that R and T each refer to amplitude, not energy. Energy goes as amplitude squared, so the fraction of energy reflected is $R^2$ and transmitted is $1 - R^2$ .

For most sedimentary reflectors with |R| < 0.2, energy loss into reflection is less than 4%. This is why a seismic wave can travel deep into the subsurface, reflecting off many boundaries along the way, and still have meaningful amplitude at each one. If reflectors were "loud" (R ≈ 0.5), the wave would be almost entirely consumed by the first boundary and we would not image below the shallow section.

Finally, a convention note. We have been assuming a wave traveling downward. If the wave is traveling up (after reflection), the roles of Z₁ and Z₂ swap, and the reflection coefficient for the upcoming wave at the same boundary is –R. This is relevant for multiples and for full-wave modeling; it is a detail that will return in Part 2.

References

Sheriff, R. E., & Geldart, L. P. (1995). Exploration Seismology (2nd ed.). Cambridge University Press.
Mavko, G., Mukerji, T., & Dvorkin, J. (2009). The Rock Physics Handbook (2nd ed.). Cambridge University Press.
Bacon, M., Simm, R., & Redshaw, T. (2003). 3-D Seismic Interpretation. Cambridge University Press.
Castagna, J. P., Batzle, M. L., & Eastwood, R. L. (1985). Relationships between compressional-wave and shear-wave velocities in clastic silicate rocks. Geophysics, 50(4), 571–581.