User:Okhakimov

Two Horizontal Layers.


ic 0 - critical angle

V0 - velocity of the first layer

V1 - velocity of the second layer

h0 - thickness of the first layer

T01 - intercept


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


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


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


 * $$ h_{0} = {X_{cross_{1}} \over 2} \sqrt$$

Several Horizontal Layers.
Forward:


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


 * $$ T0_{n} = \sum_{j=0}^{n-1}$$

Inverse:


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


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


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