# Bortfeld equation

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 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