Bortfeld equation

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Yet Another Linear Approximation to the Zoeppritz equation, from Bortfeld (1961).[1]

Original formulation

[Find the paper and add this]

Basic formulation

This formulation is from the Stanford Exploration Project, dated 6/8/2002.[2]


R(\theta_\mathrm{i}) = R_0 + R_\mathrm{sh} \sin^2\theta_\mathrm{i} + R_\mathrm{P} \tan^2\theta_\mathrm{i} \sin^2\theta_\mathrm{i} \

where


R_\mathrm{P} = \frac{\Delta V_\mathrm{P}}{2 V_\mathrm{P}} \

R_0 = R_\mathrm{P} + R_\rho \

R_\rho = \frac{\Delta \rho}{2 \rho} \

R_\mathrm{sh} = \frac12 \left(\frac{\Delta V_\mathrm{P}}{V_\mathrm{P}} - k \frac{\Delta \rho}{2 \rho} - 2k \frac{\Delta V_\mathrm{S}}{V_\mathrm{S}} \right)

k = \left(\frac{2 V_\mathrm{S}}{V_\mathrm{P}}\right)^2 \

Stack-contrained form

Again, from SEP.[3] Due to Fred Herkenhoff of Chevron. The stack amplitude is given by:


S = R_0 + R_\mathrm{sh} \sin^2\theta_\mathrm{S1} + R_\mathrm{P} \tan^2\theta_\mathrm{S2} \sin^2\theta_\mathrm{S1} \

where \sin^2\theta_\mathrm{S1} and \tan^2\theta_\mathrm{S2} are the averages of sin2θ and tan2θ over the range of input angles to the stack.

R(\theta_\mathrm{i}) - S \frac{\sin^2 \theta_\mathrm{i}}{\sin^2 \theta_\mathrm{S1}} =  R_0 \left( 1 - \frac{\sin^2 \theta_\mathrm{i}}{\sin^2 \theta_\mathrm{S1}} \right) + R_\mathrm{P} \left( \tan^2 \theta_\mathrm{i} - \frac{\tan^2\theta_\mathrm{i}\sin^2 \theta_\mathrm{i}}{\sin^2 \theta_\mathrm{S1}} \right) \

According to SEP, "this form can now be inverted for the zero-offset reflectivity (RO) and the P-wave reflectivity (RP) without needing the interval velocities."

Implementation in Modelr

Implementation in Python

Bortfled's formula is implemented in bruges.

References

  1. Bortfeld, R., 1961, Approximations to the reflection and transmission coefficients of plane longitudinal and transverse waves, Geophysical Prospecting, v.9 no. 4, 485-503.
  2. Bortfeld's 3 term reflectivity equation
  3. The Stack-Constrained form

See also