# Bortfeld equation

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

## Original formulation

[Find the paper and add this]

## Basic formulation

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

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

Again, from SEP. 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.