User:Okhakimov

Two Horizontal Layers.

Two layers model.

i_c0 - critical angle
V₀ - velocity of the first layer
V₁ - velocity of the second layer
h₀ - thickness of the first layer
T0₁ - intercept

$i_{c_{0}}=asin\left({V_{0} \over V_{1}}\right)$

$T={2h_{0}cos(i_{c_{0}}) \over V_{0}}+{X \over V_{1}}=T0_{1}+{X \over V_{1}}$

$h_{0}={T0_{1}V_{0} \over 2cos(i_{c})}$

$h_{0}={X_{cross_{1}} \over 2}{\sqrt {V_{1}-V_{0} \over V_{1}+V_{0}}}$

Several Horizontal Layers.

Forward:

$T_{n}=T0_{n}+{X \over V_{n}}$

$T0_{n}=\sum _{j=0}^{n-1}{2h_{j}cos(i_{j}) \over V_{j}}$

Inverse:

$h_{n}={V_{n} \over cos(i_{c_{n}})}\left({T0_{n+1} \over 2}-\sum _{j=0}^{n-1}{h_{j}{\sqrt {{1 \over V_{j}^{2}}-{1 \over V_{j+1}^{2}}}}}\right)$

$T0_{n}=T_{cross_{n}}-{X_{cross_{n}} \over V_{n}}$

$h_{n}={V_{n} \over cos(i_{c_{n}})}\left({\left(T_{cross_{n+1}}-{X_{cross_{n+1}} \over V_{n+1}}\right) \over 2}-\sum _{j=0}^{n-1}{h_{j}{\sqrt {{1 \over V_{j}^{2}}-{1 \over V_{j+1}^{2}}}}}\right)$