# 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]

${\displaystyle R(\theta _{\mathrm {i} })=R_{0}+R_{\mathrm {sh} }\sin ^{2}\theta _{\mathrm {i} }+R_{\mathrm {P} }\tan ^{2}\theta _{\mathrm {i} }\sin ^{2}\theta _{\mathrm {i} }\ }$

where

${\displaystyle R_{\mathrm {P} }={\frac {\Delta V_{\mathrm {P} }}{2V_{\mathrm {P} }}}\ }$
${\displaystyle R_{0}=R_{\mathrm {P} }+R_{\rho }\ }$
${\displaystyle R_{\rho }={\frac {\Delta \rho }{2\rho }}\ }$
${\displaystyle R_{\mathrm {sh} }={\frac {1}{2}}\left({\frac {\Delta V_{\mathrm {P} }}{V_{\mathrm {P} }}}-k{\frac {\Delta \rho }{2\rho }}-2k{\frac {\Delta V_{\mathrm {S} }}{V_{\mathrm {S} }}}\right)}$
${\displaystyle k=\left({\frac {2V_{\mathrm {S} }}{V_{\mathrm {P} }}}\right)^{2}\ }$

## Stack-contrained form

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

${\displaystyle S=R_{0}+R_{\mathrm {sh} }\sin ^{2}\theta _{\mathrm {S1} }+R_{\mathrm {P} }\tan ^{2}\theta _{\mathrm {S2} }\sin ^{2}\theta _{\mathrm {S1} }\ }$

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

${\displaystyle R(\theta _{\mathrm {i} })-S{\frac {\sin ^{2}\theta _{\mathrm {i} }}{\sin ^{2}\theta _{\mathrm {S1} }}}=R_{0}\left(1-{\frac {\sin ^{2}\theta _{\mathrm {i} }}{\sin ^{2}\theta _{\mathrm {S1} }}}\right)+R_{\mathrm {P} }\left(\tan ^{2}\theta _{\mathrm {i} }-{\frac {\tan ^{2}\theta _{\mathrm {i} }\sin ^{2}\theta _{\mathrm {i} }}{\sin ^{2}\theta _{\mathrm {S1} }}}\right)\ }$

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