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 (heavy and stiff) has a high $Z$ . A material that is light and slow (loose, compressible) has a low $Z$ .

For seismic work, the working definition is:

$Z = \rho V_{\mathrm P}$

where $\rho$ is bulk density (kg/m³ or g/cm³) and $V_{\mathrm P}$ is P-wave velocity (m/s). In SI units $Z$ is in kg/(m²·s); log displays use the mixed unit g/cm³·m/s, which is 1000 times larger, so its numbers are 1000 times smaller: a shale at 2.40 g/cm³ and 2700 m/s has $Z$ = 6480 on the log, or $6.48 \times 10^{6}$ kg/(m²·s). The numerical value matters less than the contrast across boundaries.

Sea water and sedimentary rocks span more than an order of magnitude in acoustic impedance (in g/cm³·m/s):

Sea water and very soft mud: about 1500 to 3000
Unconsolidated shallow sediments, and many gas sands: about 3000 to 5000
Sands and shales at moderate burial: about 5000 to 9000
Well-cemented sandstones, most carbonates, and halite: about 9000 to 13 000
Tight carbonates, dolomite, anhydrite, hard volcanic rock: about 13 000 to 20 000

Gas lowers a sand's impedance, often into the unconsolidated range. Halite is the surprise among the evaporites: at about 2.16 g/cm³ and 4500 m/s its impedance is near 9700, no higher than a cemented sandstone, so the top of salt is bright only where it lies under soft young shale. Anhydrite, at nearly 3 g/cm³ and 6000 m/s, really is at the top of the scale. The point of listing these is to calibrate your intuition: an impedance of 2800 in a buried section is unusual (water-like, or a very soft gas sand), and one of 16 000 is 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 the incident pressure 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:

Positive $R$ : impedance increases going down into the lower layer. The reflected pulse has the same polarity as the incident pulse, and on a display in SEG normal polarity it appears as a peak.
Negative $R$ : impedance decreases going down. The reflected pulse is flipped in polarity, and in SEG normal polarity it appears as a trough.
$R = 0$ : no impedance contrast, no reflection. The boundary is invisible to seismic.

In Figure 1.1 a shale sits on a sand. Change the rock below the boundary, its velocity and its density, and read the sign and the size of what comes back.

The figure opens on a real boundary that almost vanishes. The sand is faster than the shale but lighter, their impedances are 6480 and 6600, and $R = +0.009$ : a peak too weak to see through noise. Raise the sand's velocity to 3600 m/s, as cement would, and it reaches $R = +0.100$ , a typical peak. Fill it with gas instead (2.05 g/cm³, 2300 m/s) and impedance falls: $R = -0.158$ and the pulse comes back flipped, a strong trough. A strong trough at the top of a sand under shale is the classic bright spot. Exercise 3 makes the opposite point: at 2.16 g/cm³ the sand's impedance equals the shale's and the boundary disappears, although the two rocks differ in both density and velocity.

Where does this formula come from? Pressure and particle velocity must be continuous across the boundary (mechanical continuity). For a pressure wave, whose particle velocity is its pressure divided by $Z$ , the two conditions are $1 + R = T$ and $(1 - R)/Z_1 = T/Z_2$ , where $T$ is the transmitted pressure amplitude; solving them 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:

$|R|$ under 0.05: weak, may be lost in noise
$|R|$ from 0.05 to 0.15: a typical sedimentary reflector
$|R|$ from 0.15 to 0.25: strong, clearly visible, a key interpretation marker
$|R|$ over 0.25: very bright. Examples: the sea floor, shale on tight carbonate, top of salt under soft young shale, an exceptionally soft gas sand

A reflection coefficient of 0.30 means 30 % of the incident pressure amplitude comes back, but only 9 % of the energy. An $|R|$ of 0.5 is extraordinary: a hard sea floor, where water meets rock, or an exceptionally soft gas sand under hard rock. Even water over soft mud, exercise 5 in the figure, reaches only +0.329. 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 reflection coefficient, other things being roughly equal, and since $R \approx \Delta Z / (2\bar{Z})$ for small contrasts, roughly twice the relative impedance contrast. That comparative reasoning is the bread-and-butter of qualitative interpretation.

Transmission and energy conservation

Some of the wave reflects and the rest passes into the lower layer. Pressure must match on both sides, so the transmitted pressure amplitude is $T = 1 + R = 2Z_2/(Z_1 + Z_2)$ , which exceeds 1 whenever impedance rises: shale on tight carbonate transmits 1.335 times the incident pressure. No energy is made, because a wave's energy flux is its pressure squared divided by its impedance, so the stiffer rock carries more pressure for the same energy. Amplitude is not conserved across the boundary; energy is. The fraction of energy reflected is $R^2$ and the fraction transmitted is $\tfrac{Z_1}{Z_2}T^2 = 1 - R^2$ , and plate (d) of the figure shows them adding to exactly 1. (Measured as particle velocity instead of pressure, the coefficients are $-R$ and $1 - R$ : a transmission coefficient means something only with its convention stated.)

For most sedimentary reflectors $|R|$ is under 0.2, so less than 4 % of the energy reflects. 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" ( $|R| \approx 0.5$ ), each would send back a quarter of the energy, and after a handful of them almost nothing would reach the deeper section.

Finally, a convention note. We have been assuming a wave traveling downward. If the wave is traveling up (after reflection), the roles of $Z_1$ and $Z_2$ swap, and the reflection coefficient for the upgoing 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.