# Impedance, Reflectivity, & Amplitude

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Impedance, Reflectivity, & Amplitude

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Frequencies, Periods, Wavelengths, and Wavenumbers

– Period ‘T’ – is the time of repetition of a periodic wave, the time for a wave crest to travel one wavelength

– Frequency ‘f’ - is the repetition rate of the periodic wave, reciprocal of period (1/T)

– Wave length (λ) – the distance between successive similar points on a periodic wave measured perpendicular to the wave front

– Wave number - is the reciprocal of wavelength

– Amplitude – maximum displacement from equilibrium

λ

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Theory of Seismic Waves – Hook’s Law

• Hooke’s law: relates stress to strain.

• In its simplist form. In an elastic continuum we must specify: which stress, and which strain ….

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Stress, Strain • Stress and Strain

– When external forces are applied to a body, balanced internal forces are setup. Stress is a measure of these balanced internal forces.

– A body subjected to stress undergoes a change of shape/size known as strain.

– The linear relationship between stress and strain in the elastic field is specified for any material by its various elastic moduli. Stress-Strain Curve for a Solid

Body

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Elastic Moduli – Young’s Modulus – This modulus represents

the uni-axial stress -strain proportionality constant in the case of a simple dilatation or compression.

– A rod of initial length ‘l’ and cross-sectional area A is extended by ∆l by a stretching force F to its end faces.

– This has units of stress (force per unit area), or Pascals [Pa]

– Typical rock values 10 – 200 GPa

Young’s modulus

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Elastic Moduli – Bulk Modulus – This is the stress-strain

proportionality constant in the case of a simple compressive stress (hydrostatic pressure) which is uniformly acting on a certain body.

– When a pressure P is applied to a volume V, the ratio of stress to volume strain gives the Bulk modulus.

– For fluids (zero Young’s modulus, zero shear modulus), we measure the “bulk modulus”

– This has units of stress (force per unit area), or Pascals [Pa]. Water, for example, has a bulk modulus of 2.1 GPa

Bulk modulus

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Elastic Moduli - Shear Modulus

• This modulus is defined as the stress-strain proportionality constant in the case of simple shearing (or tangential) stress.

Shear modulus

Also has units of stress (force per unit area), or Pascals [Pa]

For all elastic material

For fluids, both

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Elastic Moduli – Poisson’s Ratio – The dilatation (or

compression) occurring in the case of a simple tensile (or compressive) stress, is normally accompanied by contraction (or expansion) in the direction perpendicular to the strain direction.

– The ratio of the lateral contraction (or expansion) to the longitudinal elongation (or compression) is called Poisson's Ratio

Poisson’s ratio

d d-∆d

σ = (∆d/d) / (∆l/l)

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Elastic Moduli – Poisson’s Ratio

– This is a dimensionless quantity – For most elastic solids, the value of σ is about 0.25.

The values range from 0.05 for very hard, rigid rocks to about 0.45 for soft poorly consolidated material (Telford, et al., 1976).

– For liquids which have no rigidity (that is μ = 0), σ attains its maximum possible value which is 0.5.

Elastic Properties

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Wave Propagation - Compressional Wave – Body waves travel

through the internal volume of an elastic solid and are of two types.

– Compressional waves (longitudinal, primary, or P- waves) propagate by compression and dilational uniaxial strains in the direction of wave travel

– Particle motion is in the direction of wave propagation

Velocity of P-waves is:

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Wave Propagation- Shear Wave – Shear waves

(transverse, secondary, or S-waves) propagate by a pure shear strain in a direction perpendicular to the direction of wave travel

– Particle motion is at right angles to the direction of wave propagation

Velocity of S-waves is:

Vp/Vs

• The ratio Vp/Vs is defined in terms of Poisson's ratio (σ) and is given by:

Poisson’s ratio

Vs/Vp – Indicator of Lithology

Relation between S- and P-wave velocities for various lithologies.

(a) Cross-plot of laboratory measurements (after Pickett, 1963).

(b) Use of S- and P-wave velocity as an indicator of lithology

Poisson’s Ratio

Examples of P-wave velocity, Poisson's ratio and VP ⁄ VS ratio for various lithologies

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Energy Partitioning • Seismic waves will be reflected at interfaces with “discontinuities” in elastic

properties • A new ray emerges, heading back to the surface • Energy is “partitioned” – some energy is reflected, some energy is

transmitted, according to

(where Z = vρ is the “seismic impedance”)

Note that the value of R can be negative (implies a polarity reversal) These relationships are only valid for normal incidence (i.e., zero offset)

Reflection and Transmission coefficients

(where Z = vρ is the “seismic impedance”)

• Note that the value of R can be negative (implies a polarity reversal)

• These relationships are only valid for normal incidence (i.e., zero offset)

Reflection Coefficient

Transmission coefficient

Partitioning at an Interface

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In Figure, the angles for incident, reflected and transmitted rays synchronous at the boundary are related according to Snell’s law by:

where P1V = P-wave velocity in medium 1, P2V = P-wave velocity in medium 2; S1V = S-wave velocity in medium 1; S2V = S-wave velocity in medium 2; θ1=incident P-wave angle, θ2=transmitted P-wave angle, φ1= reflected S-wave angle, φ2= transmitted S-wave angle, and p is the ray parameter.

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Partitioning of Energy – Non Normal Incidence

Bortfeld, R., 1961, Approximations to the reflection and transmission coefficients of plane longitudinal and transverse waves, Geophysical Prospecting, v.9 no. 4, 485-503.

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Amplitude Versus Angle

• Results for various velocity and density ratios and constant Poisson’s ratios of 0.2 and 0.3.

• Angle of incidence has only minor effects on P-wave reflection coefficients over propagation angles commonly used in reflection seismology.

• This is a basic principle upon which conventional common-depth-point (CDP) reflection seismology relies.

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Amplitude Versus Angle • Figure shows P-wave reflection

coefficients from an interface, with the incident medium having a higher Poisson’s ratio than the underlying medium.

• The solid curves represent a contrast in Poisson’s ratio of 0.4 to 0.1, while the dashed curves represent a contrast of 0.3 to 0.1

• These curves show that if Poisson’s ratio decreases going into the underlying medium, the reflection coefficient decreases algebraically with increasing angle of incidence.

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Amplitude Versus Angle

• In this figure the Poisson’s ratio increases going from the incident medium into the underlying medium.

• In this case, the reflection coefficients increase algebraically with increasing angle of incidence.

• Negative reflection coefficients may reverse polarity, and positive reflection coefficients increase in magnitude with increasing angle of incidence.

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List of Measured Poisson’s Ratios

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Synthetic Model Response

Three-layer hypothetical gas sand model.

Reflection Strength From Interfaces – Non-zero offset

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RP, RS, TP, and TS, are the reflected P, reflected S, transmitted P, and transmitted S-wave amplitude coefficients, respectively. Inverting the matrix form of the Zoeppritz equations give the coefficients as a function of angle.

Zoeppritz (1919) derived the particle motion amplitudes of the reflected and transmitted waves using the conservation of stress and displacement across the interface, which yields four equations with four unknowns:

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Aki & Richard’s Approximation

• Aki and Richards (1980) derived a form of approximation simply parameterized in terms of the changes in density, P-wave velocity, and S-wave velocity across the interface:

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Shuey’s Approximation

• By simplifying the Zoeppritz equations, Shuey (1985) presented another form of the Aki and Richards (1980) approximation,

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Shuey’s Approximation

• The quantity A0 , specifies the variation of R(θ) in the approximation range 0 <θ < 30° for the case of no contrast in Poisson’s ratio.

• The first term gives the amplitude at normal incidence, the second term characterizes R(θ) at intermediate angles, and the third term describes the approach to the critical angle.

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AVO Indicators

• Following Shuey (1985), the P-wave reflection coefficient as a function of angle of incidence RPP(θ) may be expressed as:

• where A is the normal incidence P-wave reflection coefficient and the AVO gradient (slope) is given by

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AVO Indicators

• Reflection coefficient definitions:

• Dependence on pore fluid content is large for RP , small for RS , and large for RP - RS.

• The lithology and porosity dependence is large for both RP and RS.

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AVO Indicators

• For shale over brine-sand reflections, the average RP - RS tends to be near zero and relatively invariant with depth.

• Irrespective of gas-sand impedance, RP - RS is always negative for shale over reservoir quality gas-sand reflections and more negative than for the corresponding brine-sand reflections.

• In comparison, the AVO product may be positive, near zero, or negative for gas-sands depending on the impedance contrast with the overlying shale.

• These measurements also verify that RP - RS is well approximated by a simple linear combination of A and B.

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AVO Indicators

• A worldwide collection of 25 sets of velocity and density measurements from adjacent shales, brine sands, and gas sands acquired with full-waveform sonic, dipole sonic, and conventional well logging devices and/or in the laboratory are used in the study.

• These data provide values for theoretical shale over brine-sand, and shale over gas-sand

• P-wave and S-wave normal-incidence reflection coefficients (RP and RS,

• AVO intercepts (A), AVO gradients (B), • the AVO indicators RP - RS (reflection coefficient

difference), • and A * B (AVO product).

• The reflection coefficient difference is found to be a more universal indicator than the AVO product in clastic stratigraphic intervals.

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AVO Indicators

• Thus, in clastic sections, nonpay RP - RS exhibits a fairly constant mean value and provides a well behaved background against which gas-sand related RP - RS clearly stands out.

• In addition, the effect of gas is always to make RP - RS more negative.

Rutherford, S. R., and Williams, R. H., 1989, Amplitude-versus offset variations in gas sands: Geophysics, 54, 680-688. Shuey, R. T., 1985, A simplification of the Zoeppritz equations: Geophysics, 50, 609-614. Smith, G. C., and Gidlow, P. M., 1987, Weighted stacking for rock property estimation and detection of gas: Geophys. Prosp., 35, 993-1014. Swan, H. W., 1993, Properties of direct AVO hydrocarbon indicators, in Castagna, J. P., and Backus, M. M., Eds., Offset dependent reflectivity: Theory and practice of AVO analysis: Soc. Expl. Geophys.

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AVO Indicators

• Following Wiggins et al. (1983) we have

• which is exact when Vp/Vs = 2. Thus, assuming appropriate measurement and calibration, (A +B)/2 should be an excellent hydrocarbon indicator in clastic sections.

• In practice, one would apply a time and space varying offset gain function to “zero out” mean (A + B)/2 in known non-pay intervals. Wiggins, R., Kenny, G. S., and McClure, C. D., 1983, A method for determining and displaying the

shear-velocity reflectivities of a geologic formation: European Patent Application 0113944.

AVO Indicators

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Average Gulf Coast RP - RS for brine sands and gas sands. The solid line is a second-order logarithmic polynomial fit to the gas-sand data.

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AVO Indicators

• P-wave reflection coefficients for 25 sets of shale, brine-sand, and gas-sand VP, VS and ρ measurements.

• The measurements represent a worldwide sampling of data, but no attempt was made to randomize the collection other than to ensure that Class I and Class II gas sands were represented.

• The reflection coefficients are for shale over brine sand (square) and shale over gas sand (plus sign).

• (b) The AVO gradients corresponding to the reflection coefficients in (a).

AVO Indicators - A * B

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A * B corresponding to the reflection coefficients in previous slide.

AVO Indicators - RP - RS

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RP - RS corresponding to the reflection coefficients in

AVO Indicators - RP - RS versus (A + B)/2

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RP - RS versus (A + B)/2 for all 25 sets of measurements.

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AVO Indicators - (A + B)/2

(A + B)/2 corresponding to the reflection coefficients.

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Castagna’s Mud Rock Line

• In-situ sonic and field seismic measurements in mudrocks form a well-defined line given by

VP = 1.16 VS, + 1.36, where the velocities are in

km/s.

Castagna, J. P., Batzle, M. L., and Eastwood, R. L., 1985, Relationships between compressional- and shear-wave velocities in clastic silicate rocks: Geophysics, 50.

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Fluid Factor

• SMITHG, .C. and GIDLOW. P.M. 1987, Weighted Stacking for Rock Property Estimation and Detection of Gas, Geophysical Prospecting 35,993-1014.

• Fluid Factor

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AVO Indicators – Fluid Factor

• Using a smooth, representative interval velocity model (from boreholes or velocity analyses) and assuming no dip, the angle of incidence can be found as a function of time and offset by iterative ray tracing.

• In particular, the angle of incidence can be computed for each sample in a normal moveout corrected CMP gather.

• The approximated Zoeppritz equation can then be fitted to the amplitudes of all the traces at each time sample of the gather, and certain rock properties can be estimated.

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AVO Indicators – Fluid Factor

• The first term is the reflection coefficient at normal incidence, the second term must be added for intermediate angles, and the last term only becomes important at larger angles of incidence.

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AVO Indicators – Fluid Factor

• Define the ratio VIW as q, then we can consider the quantity ∆q/q, which we have called ‘ pseudo-Poisson’s ratio reflectivity ’. Since

One can construct a ∆q/q section by subtracting the ∆WIW section from the ∆VIV section. Alternatively, the weights can be subtracted to arrive at a new set of weights to give ∆q/q directly.

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AVO Indicators – Fluid Factor

• Consider the ‘mudrock line’ of Castagna et al. (1985).

• All water-bearing clastic silicates should lie close to this line.

• While the substitution of gas for water reduces the P-wave velocity it hardly affects the S-wave velocity.

• ‘Fluid factor’ is defined as

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AVO Indicators – Fluid Factor

The parameters of a model. The dashed lines are the smooth functions used in the calculation.

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AVO Indicators – Fluid Factor

A synthetic CMP gather generated from the model in previous slide using the exact Zoeppritz equations.

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AVO Indicators – Fluid Factor

The results of the four weighted stacks of the gather

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AVO Indicators – Fluid Factor

P-wave velocity reflectivity for the real example

S-wave velocity reflectivity for the real example

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AVO Indicators – Fluid Factor

‘ Pseudo-Poisson’s ratio ’ reflectivity for the real example.

‘Fluid factor’ for the real example.