Determination of Dips and Depths of Near Surface Layers by Radon Transform

Abstract

The refracted arrivals on seismic shot records have long been recognized as an efficient tool for the computation of detailed near-surface information. In this paper, a new concept of refraction static, which is based on the Radon transform and avoids the tedious process of picking first arrival times, is proposed. This method is particularly suitable when a rough near-surface problem necessitates the utilization of numerous shallow refraction data for the one reflector case. Quasi-linearity of refractors and a constant velocity medium are assumed within the shooting range. Synthetic and real cases have been tested to evaluate the performance of the method. The result is revealed to be satisfactory. Comparison of the synthetic model with the results obtained through the Radon transform reveals a very good accuracy for the proposed method.

Appendices

Appendix A

Three layers model shown in Fig. 5a will be used as a basis to formulate general equations that represent the travel times t _OABCDE , $t_{{OA^{\prime}B^{\prime}C^{\prime}D^{\prime}E^{\prime}}}$ and to calculate relevant velocities and thickness terms.

The same statement about the sign of the dip angle remains valid as explained in the main text so that we can handle the problem no matter what is the dip orientation of the interfaces.

Calculations related to +x direction:

To evaluate t _ABCDEF , extract the appropriate velocity and thickness terms connected to the trave ltime, and we establish first, referring to Fig. 5a, the following set of relations

$$ OE = x. $$

$$ AD = x\left( {\frac{{\cos (\gamma_{2,1} - \theta_{1} )}}{{\cos \gamma_{2,1} }}} \right) - Z_{1} \left( {\frac{{\sin (\varphi_{2,1} + \gamma_{2,1} )}}{{\cos \varphi_{2,1} \cos \gamma_{2,1} }}} \right). $$

$$ BC = AD\left( {\frac{{\cos (i_{2} - \theta_{2} + \theta_{1} )}}{{\cos i_{2} }}} \right) - 2Z_{2,2} \tan i_{2} . $$

$$ OA = \frac{{Z_{1} }}{{\cos \varphi_{2,1} }},\text{ }AB = \frac{{Z_{2,2} }}{{\cos \varphi_{2,2} }}, $$

$$ CD = \frac{{z^{\prime}_{2,2} }}{{\cos \gamma_{2,2} }},\text{ }DE = \frac{{z_{1} }}{{\cos \gamma_{2,1} }}, $$

$$ z_{1} = Z_{1} - x\sin \theta_{1} ,\text{ }v_{2} = \frac{{v_{1} }}{{\sin i_{2} }}, $$

$$ z_{2,2} = Z_{2,2} - AD\sin \, \left( {\theta_{2} - \theta_{1} } \right),\text{ }i_{2} = \varphi_{2,2} = \gamma_{2,2} . $$

(A-1)

Using the above expressions the total travel time t ⁺ _x2 can be formulated as

$$ t_{x2}^{ + } = t_{OABCDE} = \frac{OA}{{v_{\text{o}} }} + \frac{AB}{{v_{1} }} + \frac{BC}{{v_{2} }} + \frac{CD}{{v_{1} }} + \frac{DE}{{v_{\text{o}} }}, $$

$$ t_{x2}^{ + } = \frac{{Z_{1} }}{{v_{\text{o}} \cos \varphi_{2,1} }} + \frac{{Z_{2,2} }}{{v_{1} \cos \varphi_{2,2} }} + \frac{BC}{{v_{2} }} + \frac{{z_{2,2} }}{{v_{1} \cos \gamma_{2,2} }} + \frac{{z_{1} }}{{v_{\text{o}} \cos \gamma_{2,1} }}, $$

$$ t_{x2}^{ + } = \frac{BC}{{v_{2} }} + \frac{{2Z_{2,2} - AD\sin (\theta_{2} - \theta_{1} )}}{{v_{1} \cos i_{2} }} + \frac{{Z_{1} }}{{v_{o} \cos \varphi_{2,1} }} + \frac{{Z_{1} - x\sin \theta_{1} }}{{v_{o} \cos \gamma_{2,1} }}, $$

$$ t_{x2}^{ + } = AD\left( {\frac{{\cos (i_{2} - \theta_{2} + \theta_{1} )}}{{v_{2} \cos i_{2} }}} \right) - \frac{{2Z_{2,2} \tan i_{2} }}{{v_{2} }} + \frac{{2Z_{2,2} - AD\sin (\theta_{2} - \theta_{1} )}}{{v_{1} \cos i_{2} }} + \frac{{Z_{1} }}{{v_{\text{o}} \cos \varphi_{2,1} }} + \frac{{Z_{1} - x\sin \theta_{1} }}{{v_{\text{o}} \cos \gamma_{2,1} }}, $$

$$ t_{x2}^{ + } = AD\left( {\frac{{\sin i_{2} \cos (i_{2} - \theta_{2} + \theta_{1} ) - \sin (\theta_{2} - \theta_{1} )}}{{v_{1} \cos i_{2} }}} \right) - \frac{{x\sin \theta_{1} }}{{v_{\text{o}} \cos \gamma_{2,1} }} + \frac{{2Z_{2,2} \cos i_{2} }}{{v_{1} }} + Z_{1} \left( {\frac{{\cos \varphi_{2,1} + \cos \gamma_{2,1} }}{{v_{\text{o}} \cos \varphi_{2,1} \cos \gamma_{2,1} }}} \right), $$

(A-2)

$$ t_{x2}^{ + } = P + Q + \frac{{2Z_{2,2} \cos i_{2} }}{{v_{1} }}, $$

(A-3)

where

$$ P = \frac{{x\cos (\gamma_{2,1} - \theta_{1} )}}{{\cos \gamma_{2,1} }}\left( {\frac{{\sin i_{2} \cos (i_{2} - \theta_{2} + \theta_{1} ) - \sin (\theta_{2} - \theta_{1} )}}{{v_{1} \cos i_{2} }}} \right) - \frac{{x\sin \theta_{1} }}{{v_{o} \cos \gamma_{2,1} }}, $$

(A-4)

$$ Q = - Z_{1} \frac{{\sin (\varphi_{2,1} + \gamma_{2,1} )}}{{\cos \varphi_{2,1} \cos \gamma_{2,1} }}\left( {\frac{{\sin i_{2} \cos (i_{2} - \theta_{2} + \theta_{1} ) - \sin (\theta_{2} - \theta_{1} )}}{{v_{1} \cos i_{2} }}} \right) + Z_{1} \left( {\frac{{\cos \varphi_{2,1} + \cos \gamma_{2,1} }}{{v_{\text{o}} \cos \varphi_{2,1} \cos \gamma_{2,1} }}} \right). $$

(A-5)

We, now, try to simplify and put in more compact form expressions (A-4) and (A-5)

$$ P = \frac{{x\cos (\gamma_{2,1} - \theta_{1} )}}{{v_{1} \cos i_{2} \cos \gamma_{2,1} }}\left( {\sin i_{2} \left( {\cos i_{2} \cos \left( {\theta_{2} - \theta_{1} } \right) + \sin i_{2} \sin \left( {\theta_{2} - \theta_{1} } \right)} \right) - \sin \left( {\theta_{2} - \theta_{1} } \right)} \right) - \frac{{x\sin \theta_{1} }}{{v_{o} \cos \gamma_{2,1} }}, $$

$$ P = \frac{{x\cos (\gamma_{2,1} - \theta_{1} )}}{{v_{1} \cos i_{2} \cos \gamma_{2,1} }}\left( {\cos i_{2} \sin \left( {i_{2} - \theta_{2} + \theta_{1} } \right)} \right) - \frac{{x\sin \theta_{1} }}{{v_{o} \cos \gamma_{2,1} }}, $$

$$ P = \frac{{x\cos (\gamma_{2,1} - \theta_{1} )}}{{v_{1} \cos \gamma_{2,1} }}\sin (i_{2} - \theta_{2} + \theta_{1} ) - \frac{{x\sin \theta_{1} }}{{v_{o} \cos \gamma_{2,1} }}. $$

(A-6)

Using Snell’s law

$$ \frac{{v_{\text{o}} }}{{\sin \varphi_{2,1} }} = \frac{{v_{1} }}{{\sin (\alpha_{2,1} )}} = \frac{{v_{1} }}{{\sin (i_{2} + \theta_{2} - \theta_{1} )}},\frac{{v_{\text{o}} }}{{\sin \gamma_{2,1} }} = \frac{{v_{1} }}{{\sin (\beta_{2,1} )}} = \frac{{v_{1} }}{{\sin (i_{2} - \theta_{2} + \theta_{1} )},} $$

(A-7)

$$ v_{1} = \frac{{v_{\text{o}} \sin (i_{2} + \theta_{2} - \theta_{1} )}}{{\sin \varphi_{2,1} }}\,\text{ }{\text{or}}\,\text{ }v_{1} = \frac{{v_{\text{o}} \sin (i_{2} - \theta_{2} + \theta_{1} )}}{{\sin \gamma_{2,1} }} $$

(A-8)

and putting (A-8) into (A-6) gives

$$ P = \frac{{x\cos (\gamma_{2,1} - \theta_{1} )}}{{\cos \gamma_{2,1} \frac{{v_{\text{o}} \sin (i_{2} - \theta_{2} + \theta_{1} )}}{{\sin \gamma_{2,1} }}}}\sin (i_{2} - \theta_{2} + \theta_{1} ) - \frac{{x\sin \theta_{1} }}{{v_{\text{o}} \cos \gamma_{2,1} }}, $$

$$ P = \frac{{x\cos (\gamma_{2,1} - \theta_{1} )\sin \beta_{2,1} }}{{v_{\text{o}} \cos \gamma_{2,1} }} - \frac{{x\sin \theta_{1} }}{{v_{\text{o}} \cos \gamma_{2,1} }}, $$

$$ P = \frac{{x\cos (\gamma_{2,1} - \theta_{1} )\sin \gamma_{2,1} - x\sin \theta_{1} }}{{v_{\text{o}} \cos \gamma_{2,1} }}, $$

$$ P = \frac{{x\sin \gamma_{2,1} \left( {\cos \gamma_{2,1} \cos \theta_{1} + \sin \gamma_{2,1} \sin \theta_{1} } \right) - x\sin \theta_{1} }}{{v_{\text{o}} \cos \gamma_{2,1} }}, $$

$$ P = \frac{{x\sin \left( {\gamma_{2,1} - \theta_{1} } \right)}}{{v_{\text{o}} }}. $$

( A-9)

A similar calculation is carried out on Q yielding

$$ Q = - Z_{1} \frac{{\sin (\alpha_{2,1} + \beta_{2,1} )}}{{\cos \alpha_{2,1} \cos \beta_{2,1} }}\left( {\frac{{\sin i_{2} \cos (i_{2} - \theta_{2} + \theta_{1} ) - \sin (\theta_{2} - \theta_{1} )}}{{v_{1} \cos i_{2} }}} \right) + Z_{1} \left( {\frac{{\cos \alpha_{2,1} + \cos \beta_{2,1} }}{{v_{\text{o}} \cos \alpha_{2,1} \cos \beta_{2,1} }}} \right), $$

$$ Q = - Z_{1} \frac{{\sin (\alpha_{2,1} + \beta_{2,1} )}}{{\cos \alpha_{2,1} \cos \beta_{2,1} }}\left( {\frac{{\sin (i_{2} - \theta_{2} + \theta_{1} )}}{{v_{1} }}} \right) + Z_{1} \left( {\frac{{\cos \alpha_{2,1} + \cos \beta_{2,1} }}{{v_{\text{o}} \cos \alpha_{2,1} \cos \beta_{2,1} }}} \right), $$

$$ v_{1} = \frac{{v_{\text{o}} \sin (i_{2} - \theta_{2} + \theta_{1} )}}{{\sin \beta_{2,1} }}, $$

$$ Q = - Z_{1} \frac{{\sin (\varphi_{2,1} + \gamma_{2,1} )}}{{\cos \varphi_{2,1} \cos \gamma_{2,1} }}\left( {\frac{{\sin (i_{2} - \theta_{2} + \theta_{1} )\sin \gamma_{2,1} }}{{v_{\text{o}} \sin (i_{2} - \theta_{2} + \theta_{1} )}}} \right) + Z_{1} \left( {\frac{{\cos \varphi_{2,1} + \cos \gamma_{2,1} }}{{v_{\text{o}} \cos \varphi_{2,1} \cos \gamma_{2,1} }}} \right), $$

$$ Q = - Z_{1} \frac{{\sin \gamma_{2,1} \sin (\varphi_{2,1} + \gamma_{2,1} )}}{{v_{\text{o}} \cos \varphi_{2,1} \cos \gamma_{2,1} }} + Z_{1} \left( {\frac{{\cos \varphi_{2,1} + \cos \gamma_{2,1} }}{{v_{\text{o}} \cos \varphi_{2,1} \cos \gamma_{2,1} }}} \right), $$

$$ Q = - Z_{1} \frac{{\sin \gamma_{2,1} \left( {\sin \varphi_{2,1} \cos \gamma_{2,1} + \cos \alpha_{2,1} \sin \gamma_{2,1} } \right)}}{{v_{\text{o}} \cos \varphi_{2,1} \cos \gamma_{2,1} }} + Z_{1} \left( {\frac{{\cos \varphi_{2,1} + \cos \gamma_{2,1} }}{{v_{\text{o}} \cos \varphi_{2,1} \cos \gamma_{2,1} }}} \right), $$

$$ Q = Z_{1} \frac{{\cos \gamma_{2,1} \left( {\cos \varphi_{2,1} \cos \gamma_{2,1} - \sin \varphi_{2,1} \sin \gamma_{2,1} } \right)}}{{v_{\text{o}} \cos \varphi_{2,1} \cos \gamma_{2,1} }} + Z_{1} \left( {\frac{{\cos \gamma_{2,1} }}{{v_{\text{o}} \cos \varphi_{2,1} \cos \gamma_{2,1} }}} \right), $$

$$ Q = Z_{1} \frac{{\cos \left( {\varphi_{2,1} + \gamma_{2,1} } \right)}}{{v_{\text{o}} \cos \varphi_{2,1} }} + Z_{1} \left( {\frac{1}{{v_{\text{o}} \cos \varphi_{2,1} }}} \right), $$

$$ Q = \frac{{Z_{1} }}{{v_{\text{o}} }}\frac{{\cos \left( {\varphi_{2,1} + \gamma_{2,1} } \right) + 1}}{{\cos \varphi_{2,1} }}. $$

(A-10)

Replacing (A-9) and (A-10) in (A-3), the travel time expression becomes

$$ t_{x}^{ + } = \frac{{x\sin \left( {\gamma_{2,1} - \theta_{1} } \right)}}{{v_{\text{o}} }} + \frac{{Z_{1} }}{{v_{\text{o}} }}\frac{{\cos \left( {\varphi_{2,1} + \gamma_{2,1} } \right) + 1}}{{\cos \varphi_{2,1} }} + \frac{{2Z_{2,2} \cos i_{2} }}{{v_{1} }}, $$

(A-11)

$$ t_{x}^{ + } = \frac{x}{{v_{r2} }} + t_{i2} , $$

(A-12)

where

$$ v_{r2} = \frac{{v_{\text{o}} }}{{\sin \left( {\gamma_{2,1} - \theta_{1} } \right)}}, $$

(A-13)

and

$$ t_{i2} = \frac{{Z_{1} }}{{v_{\text{o}} }}\frac{{\cos \left( {\varphi_{2,1} + \gamma_{2,1} } \right) + 1}}{{\cos \varphi_{2,1} }} + \frac{{2Z_{2,2} \cos i_{2} }}{{v_{1} }}. $$

(A-14)

From (A-14) we find easily the thickness Z _2,2 of the second layer at point A.

$$ Z_{2,2} = \frac{{v_{1} \left( {t_{i2} - \frac{{Z_{1} }}{{v_{\text{o}} }}\frac{{\cos \left( {\varphi_{2,1} + \gamma_{2,1} } \right) + 1}}{{\cos \varphi_{2,1} }}} \right)}}{{2\cos i_{2} }}. $$

(A-15)

In order to calculate Z _2,2 we, first, obtain θ ₁, v _o and Z ₁ as explained in the previous part of this paper.

After determining, through Radon transform, apparent velocities v _r2 and v _l2, we can write (Fig. 5a) the following expressions

$$ \varphi_{2,1} = {\text{arc }}\sin \frac{{v_{\text{o}} }}{{v_{l2} }} - \theta_{1} , $$

(A-16)

$$ \gamma_{2,1} = {\text{arc }}\sin \frac{{v_{\text{o}} }}{{v_{r2} }} + \theta_{1} , $$

(A-17)

$$ \alpha_{2,1} = {\text{arc }}\sin \left( {\frac{{v_{1} }}{{v_{\text{o}} }}\sin \varphi_{2,1} } \right), $$

(A-18)

$$ \beta_{2,1} = {\text{arc }}\sin \left( {\frac{{v_{1} }}{{v_{\text{o}} }}\sin \gamma_{2,1} } \right), $$

(A-19)

$$ \varphi_{2,1} = \alpha_{2,1} - (\theta_{2} - \theta_{1} ), $$

(A-20)

$$ \gamma_{2,1} = \beta_{2,1} + (\theta_{2} - \theta_{1} ), $$

(A-21)

$$ \theta_{2} = \frac{{\alpha_{2,1} - \beta_{2,1} }}{2} + \theta_{1} , $$

(A-22)

$$ i_{2} = \varphi_{2,2} = \gamma_{2,2} = \frac{{\alpha_{2,1} + \beta_{2,1} }}{2}. $$

( A-23)

Using (A-8) and (A-12), v ₂ is found to be as

$$ v_{2} = \frac{{v_{1} }}{{\sin i_{2} }}. $$

(A-24)

Putting (A-16) and (A-17) into (A-18) and (A-19), respectively, we find α _2,1 and β _2,1. After that, we calculate θ ₂ and i ₂ by using (A-22) and (A-23).

Evaluation of Z _2t and H ₂:

We can write (Fig. 5b) following expression

$$ Z_{2t} = ON + NW + WK. $$

(A-25)

According to the sign of sin(θ ₂ − θ ₁), NW = NA sin(θ ₂ − θ ₁) may be either negative or positive. But in both cases (A-25) holds, always, true.

In (A-25)

$$ NA = MA - MN, $$

$$ NA = Z_{1} \tan \varphi_{2,1} - Z_{1} \tan \left( {\theta_{2} - \theta_{1} } \right). $$

(A-26)

In this formula tan(θ ₂ − θ ₁) determines the sign of Z ₁ tan(θ ₂ − θ ₁) so that the length of Z ₁ tan(θ ₂ − θ ₁) should be added or subtracted from MA according to the sign of tan(θ ₂ − θ ₁), if we put

$$ ON = \frac{{Z_{1} }}{{\cos \left( {\theta_{2} - \theta_{1} } \right)}}, $$

(A-27)

and

$$ WK = Z_{2,2} . $$

(A-28)

In (A-25) we obtain

$$ \begin{gathered} Z_{2t} = \frac{{Z_{1} }}{{\cos \left( {\theta_{2} - \theta_{1} } \right)}} + Z_{1} \sin \left( {\theta_{2} - \theta_{1} } \right)\left[ {\tan \varphi_{2,1} - \tan \left( {\theta_{2} - \theta_{1} } \right)} \right] + Z_{2,2} , \hfill \\ Z_{2t} = Z_{1} \left[ {\frac{{\cos \left( {\varphi_{2,1} - \theta_{2} + \theta_{1} } \right)}}{{\cos \varphi_{2,1} }}} \right] + Z_{2,2} \hfill. \\ \end{gathered} $$

(A-29)

This formula is equivalent to that already developed by Mota (1954). But it is valid whatever is the orientation of the interfaces. In other words, whatever the sign of θ ₁ and θ ₂. Another requested quantity H ₂ is as

$$ H_{2} = \frac{{Z_{2t} }}{{\cos \theta_{2} }} = \frac{1}{{\cos \theta_{2} }}\left[ {Z_{1} \frac{{\cos \left( {\varphi_{2,1} - \theta_{2} + \theta_{1} } \right)}}{{\cos \varphi_{2,1} }} + Z_{2,2} } \right]. $$

(A-30)

Calculations related to −x direction:

The same line of reasoning, used for the +x direction, yields the following expressions.

$$ t_{x2}^{ - } = t_{{OA^{\prime}B^{\prime}C^{\prime}D^{\prime}E^{\prime}}} = \frac{{OA^{\prime}}}{{v_{\text{o}} }} + \frac{{A^{\prime}B^{\prime}}}{{v_{1} }} + \frac{{B^{\prime}C^{\prime}}}{{v_{2} }} + \frac{{C^{\prime}D^{\prime}}}{{v_{1} }} + \frac{{D^{\prime}E^{\prime}}}{{v_{\text{o}} }}, $$

(A-31)

where individual ray path lengths are

$$ OE^{\prime } = x, $$

$$ OA^{\prime} = \frac{{Z_{1} }}{{\cos \gamma_{2,1} }},\,\,A^{\prime}B^{\prime} = \frac{{Z^{\prime}_{2,2} }}{{\cos \gamma_{2,2} }}, $$

$$ C^{\prime}D^{\prime} = \frac{{z^{\prime}_{2,2} }}{{\cos \varphi_{2,2} }},\,\,D^{\prime}E^{\prime} = \frac{{z^{\prime}_{1} }}{{\cos \varphi_{2,1} }}, $$

$$ B^{\prime}C^{\prime} = A^{\prime}D^{\prime}\left( {\frac{{\cos (i_{2} + \theta_{2} - \theta_{1} )}}{{\cos i_{2} }}} \right) - 2Z^{\prime}_{2,2} \tan i_{2} , $$

(A-32)

where

$$ \begin{gathered} A^{\prime } D^{\prime } = x\cos \theta_{1} - (Z_{1} \tan \gamma_{2,1} + z^{\prime }_{1} \tan \varphi_{2,1} ), \hfill \\ A^{\prime}D^{\prime} = x\left( {\frac{{\cos (\varphi_{2,1} + \theta_{1} )}}{{\cos \varphi_{2,1} }}} \right) - Z_{1} \left( {\frac{{\sin (\varphi_{2,1} + \gamma_{2,1} )}}{{\cos \varphi_{2,1} \cos \gamma_{2,1} }}} \right). \hfill \\ \end{gathered} $$

(A-33)

Combining (A-31), (A-32), (A-33) and keeping in mind that

$$ t_{x2}^{ - } = \frac{{OA^{\prime}}}{{v_{\text{o}} }} + \frac{{A^{\prime}B^{\prime}}}{{v_{1} }} + \frac{{B^{\prime}C^{\prime}}}{{v_{2} }} + \frac{{C^{\prime}D^{\prime}}}{{v_{1} }} + \frac{{D^{\prime}E^{\prime}}}{{v_{\text{o}} }}, $$

$$ t_{x2}^{ - } = \frac{{Z_{1} }}{{v_{\text{o}} \cos \gamma_{2,1} }} + \frac{{Z^{\prime}_{2,2} }}{{v_{1} \cos \gamma_{2,2} }} + \frac{{B^{\prime}C^{\prime}}}{{v_{2} }} + \frac{{z^{\prime}_{2,2} }}{{v_{1} \cos \gamma_{2,2} }} + \frac{{z^{\prime}_{1} }}{{v_{\text{o}} \cos \gamma_{2,1} }}, $$

$$ t_{x2}^{ - } = \frac{{B^{\prime}C^{\prime}}}{{v_{2} }} + \frac{{2Z^{\prime}_{2,2} + A^{\prime}D^{\prime}\sin (\theta_{2} - \theta_{1} )}}{{v_{1} \cos i_{2} }} + \frac{{Z_{1} }}{{v_{\text{o}} \cos \gamma_{2,1} }} + \frac{{Z_{1} + x\sin \theta_{1} }}{{v_{\text{o}} \cos \varphi_{2,1} }}, $$

$$ t_{x2}^{ - } = A^{\prime}D^{\prime}\left( {\frac{{\cos (i_{2} + \theta_{2} - \theta_{1} )}}{{v_{2} \cos i_{2} }}} \right) - \frac{{2Z^{\prime}_{2,2} \tan i_{2} }}{{v_{2} }} + \frac{{2Z^{\prime}_{2,2} + A^{\prime}D^{\prime}\sin (\theta_{2} - \theta_{1} )}}{{v_{1} \cos i_{2} }} + \frac{{Z_{1} }}{{v_{\text{o}} \cos \gamma_{2,1} }} + \frac{{Z_{1} + x\sin \theta_{1} }}{{v_{\text{o}} \cos \varphi_{2,1} }}. $$

We find that

$$ t_{x2}^{ - } = A^{\prime}D^{\prime}\left( {\frac{{\sin i_{2} \cos (i_{2} + \theta_{2} - \theta_{1} ) + \sin (\theta_{2} - \theta_{1} )}}{{v_{1} \cos i_{2} }}} \right) + \frac{{2Z^{\prime}_{2,2} \cos i_{2} }}{{v_{1} }} + \frac{{Z_{1} }}{{v_{\text{o}} \cos \gamma_{2,1} }} + \frac{{Z_{1} + x\sin \theta_{1} }}{{v_{\text{o}} \cos \varphi_{2,1} }}, $$

$$ t_{x2}^{ - } = A^{\prime}D^{\prime}\left( {\frac{{\sin i_{2} \cos (i_{2} + \theta_{2} - \theta_{1} ) + \sin (\theta_{2} - \theta_{1} )}}{{v_{1} \cos i_{2} }}} \right) + \frac{{x\sin \theta_{1} }}{{v_{\text{o}} \cos \varphi_{2,1} }} + \frac{{2Z^{\prime}_{2,2} \cos i_{2} }}{{v_{1} }} + Z_{1} \left( {\frac{{\cos \varphi_{2,1} + \cos \gamma_{2,1} }}{{v_{\text{o}} \cos \varphi_{2,1} \cos \gamma_{2,1} }}} \right), $$

$$ t_{x2}^{ - } = A^{\prime}D^{\prime}\left( {\frac{{\sin (i_{2} + \theta_{2} - \theta_{1} )}}{{v_{1} }}} \right) + \frac{{x\sin \theta_{1} }}{{v_{\text{o}} \cos \varphi_{2,1} }} + \frac{{2Z^{\prime}_{2,2} \cos i_{2} }}{{v_{1} }} + Z_{1} \left( {\frac{{\cos \varphi_{2,1} + \cos \gamma_{2,1} }}{{v_{\text{o}} \cos \varphi_{2,1} \cos \gamma_{2,1} }}} \right), $$

$$ \begin{gathered} t_{x2}^{ - } = \left( {\frac{{\sin (i_{2} + \theta_{2} - \theta_{1} )}}{{v_{1} }}} \right)\left( {x\left( {\frac{{\cos (\varphi_{2,1} + \theta_{1} )}}{{\cos \varphi_{2,1} }}} \right) - Z_{1} \left( {\frac{{\sin (\varphi_{2,1} + \gamma_{2,1} )}}{{\cos \varphi_{2,1} \cos \gamma_{2,1} }}} \right)} \right) + \frac{{x\sin \theta_{1} }}{{v_{\text{o}} \cos \varphi_{2,1} }} + \frac{{2Z^{\prime}_{2,2} \cos i_{2} }}{{v_{1} }} + Z_{1} \left( {\frac{{\cos \varphi_{2,1} + \cos \gamma_{2,1} }}{{v_{\text{o}} \cos \varphi_{2,1} \cos \gamma_{2,1} }}} \right) \hfill \\ t_{x2}^{ - } = x\left( {\frac{{\sin (i_{2} + \theta_{2} - \theta_{1} )\cos (\varphi_{2,1} + \theta_{1} )}}{{v_{1} \cos \varphi_{2,1} }}} \right) + \frac{{x\sin \theta_{1} }}{{v_{\text{o}} \cos \varphi_{2,1} }} - Z_{1} \left( {\frac{{\sin (i_{2} + \theta_{2} - \theta_{1} )\sin (\varphi_{2,1} + \gamma_{2,1} )}}{{v_{1} \cos \varphi_{2,1} \cos \gamma_{2,1} }}} \right) + Z_{1} \left( {\frac{{\cos \varphi_{2,1} + \cos \gamma_{2,1} }}{{v_{\text{o}} \cos \varphi_{2,1} \cos \gamma_{2,1} }}} \right) + \frac{{2Z^{\prime}_{2,2} \cos i_{2} }}{{v_{1} }}. \hfill \\ \end{gathered} $$

$$ t_{x}^{ - } = P^{\prime} + Q^{\prime} + \frac{{2Z^{\prime}_{2,2} \cos i_{2} }}{{v_{1} }}, $$

(A-34)

where

$$ P^{\prime} = x\left( {\frac{{\sin (i_{2} + \theta_{2} - \theta_{1} )\cos (\alpha_{2,1} + \theta_{1} )}}{{v_{1} \cos \varphi_{2,1} }}} \right) + \frac{{x\sin \theta_{1} }}{{v_{\text{o}} \cos \varphi_{2,1} }}, $$

$$ Q^{\prime} = - Z_{1} \left( {\frac{{\sin (i_{2} + \theta_{2} - \theta_{1} )\sin (\varphi_{2,1} + \gamma_{2,1} )}}{{v_{1} \cos \varphi_{2,1} \cos \gamma_{2,1} }}} \right) + Z_{1} \left( {\frac{{\cos \varphi_{2,1} + \cos \gamma_{2,1} }}{{v_{o} \cos \varphi_{2,1} \cos \gamma_{2,1} }}} \right), $$

(A-35)

and, finally, we find

$$ P^{\prime} = \frac{{x\sin \left( {\varphi_{2,1} + \theta_{1} } \right)}}{{v_{\text{o}} }}. $$

$$ Q^{\prime} = \frac{{Z_{1} }}{{v_{\text{o}} }}\frac{{\cos \left( {\varphi_{2,1} + \gamma_{2,1} } \right) + 1}}{{\cos \gamma_{2,1} }}. $$

$$ t_{x2}^{ - } = \frac{{x\sin \left( {\varphi_{2,1} + \theta_{1} } \right)}}{{v_{\text{o}} }} + \frac{{Z_{1} }}{{v_{\text{o}} }}\frac{{\cos \left( {\varphi_{2,1} + \gamma_{2,1} } \right) + 1}}{{\cos \gamma_{2,1} }} + \frac{{2Z^{\prime}_{2,2} \cos i_{2} }}{{v_{1} }}. $$

$$ t_{x2}^{ - } = \frac{x}{{v_{l2} }} + t^{\prime}_{i2} , $$

(A-36)

where

$$ v_{l2} = \frac{{v_{\text{o}} }}{{\sin \left( {\varphi_{2,1} + \theta_{1} } \right)}}, $$

(A-37)

and

$$ t^{\prime}_{i2} = \frac{{Z_{1} }}{{v_{\text{o}} }}\frac{{\cos \left( {\varphi_{2,1} + \gamma_{2,1} } \right) + 1}}{{\cos \gamma_{2,1} }} + \frac{{2Z^{\prime}_{2,2} \cos i_{2} }}{{v_{1} }}. $$

(A-38)

From (A-38) $Z_{2,2}^{{\prime }}$ is found to be

$$ Z^{\prime}_{2,2} = \frac{{v_{1} \left( {t^{\prime}_{i2} - \frac{{Z_{1} }}{{v_{\text{o}} }}\frac{{\cos \left( {\varphi_{2,1} + \gamma_{2,1} } \right) + 1}}{{\cos \gamma_{2,1} }}} \right)}}{{2\cos i_{2} }}. $$

(A-39)

Evaluation of Z _2t and H ₂:

We can write (Fig. 5c) following expression

$$ Z_{2t} = ON + NK, $$

(A-40)

where

$$ ON = \frac{{Z_{1} }}{{\cos \left( {\theta_{2} - \theta_{1} } \right)}}, $$

(A-41)

and

$$ NK = Z^{\prime}_{2,2} - A^{\prime}N\sin (\theta_{2} - \theta_{1} ). $$

(A-42)

If we put

$$ Z_{2t} = \frac{{Z_{1} }}{{\cos \left( {\theta_{2} - \theta_{1} } \right)}} + Z^{\prime}_{2,2} - A^{\prime}N\sin (\theta_{2} - \theta_{1} ). $$

(A-43)

A ^′ N sin(θ ₂ − θ ₁) may be negative or positive according to the sign of (θ ₂ − θ ₁).

Simplification of formula (A-43) yields

$$ Z_{2t} = Z_{1} \left[ {\frac{{\cos \left( {\gamma_{2,1} + \theta_{2} - \theta_{1} } \right)}}{{\cos \gamma_{2,1} }}} \right] + Z^{\prime}_{2,2} , $$

(A-44)

and

$$ H_{2} = \frac{{Z_{2t} }}{{\cos \theta_{2} }} = \frac{1}{{\cos \theta_{2} }}\left[ {Z_{1} \frac{{\cos \left( {\gamma_{2,1} + \theta_{2} - \theta_{1} } \right)}}{{\cos \gamma_{2,1} }} + Z^{\prime}_{2,2} } \right]. $$

(A-45)

Appendix B

Demonstration of the equality $t_{i2} = t_{i2}^{\prime }$ follows:

We know from (A-14) and (A-38)

$$ t_{i2} = \frac{{Z_{1} }}{{v_{\text{o}} }}\frac{{\cos \left( {\varphi_{2,1} + \gamma_{2,1} } \right) + 1}}{{\cos \varphi_{2,1} }} + \frac{{2Z_{2,2} \cos i_{2} }}{{v_{1} }}. $$

$$ t^{\prime}_{i2} = \frac{{Z_{1} }}{{v_{\text{o}} }}\frac{{\cos \left( {\varphi_{2,1} + \gamma_{2,1} } \right) + 1}}{{\cos \gamma_{2,1} }} + \frac{{2Z^{\prime}_{2,2} \cos i_{2} }}{{v_{1} }}. $$

Let’s suppose that

$$ t_{i2} = t^{\prime}_{i2} , $$

(B-1)

$$ \frac{{Z_{1} }}{{v_{\text{o}} }}\frac{{\cos \left( {\varphi_{2,1} + \gamma_{2,1} } \right) + 1}}{{\cos \varphi_{2,1} }} + \frac{{2Z_{2,2} \cos i_{2} }}{{v_{1} }} = \frac{{Z_{1} }}{{v_{\text{o}} }}\frac{{\cos \left( {\varphi_{2,1} + \gamma_{2,1} } \right) + 1}}{{\cos \gamma_{2,1} }} + \frac{{2Z^{\prime}_{2,2} \cos i_{2} }}{{v_{1} }}, $$

$$ \frac{{Z_{1} }}{{v_{\text{o}} }}\left( {\frac{{\cos \left( {\varphi_{2,1} + \gamma_{2,1} } \right) + 1}}{{\cos \varphi_{2,1} }} - \frac{{\cos \left( {\varphi_{2,1} + \gamma_{2,1} } \right) + 1}}{{\cos \gamma_{2,1} }}} \right) = \frac{{2\cos i_{2} }}{{v_{1} }}\left( {Z^{\prime}_{2,2} - Z_{2,2} } \right), $$

$$ Z_{{_{2,2} }}^{\prime } = Z_{1} \sin \left( {\theta_{2} - \theta_{1} } \right)(\tan \varphi_{2,1} + \tan \gamma_{2,1} ) + Z_{2,2} , $$

$$ \frac{{Z_{1} }}{{v_{\text{o}} }}\left( {\frac{{\cos \left( {\varphi_{2,1} + \gamma_{2,1} } \right) + 1}}{{\cos \varphi_{2,1} }} - \frac{{\cos \left( {\varphi_{2,1} + \gamma_{2,1} } \right) + 1}}{{\cos \gamma_{2,1} }}} \right) = \frac{{2\cos i_{2} }}{{v_{1} }}Z_{1} \sin \left( {\theta_{2} - \theta_{1} } \right)\left( {\tan \varphi_{2,1} + \tan \gamma_{2,1} } \right), $$

$$ \frac{{Z_{1} \left( {\cos \left( {\varphi_{2,1} + \gamma_{2,1} } \right) + 1} \right)}}{{v_{\text{o}} }}\left( {\frac{{\cos \gamma_{2,1} - \cos \varphi_{2,1} }}{{\cos \varphi_{2,1} \cos \gamma_{2,1} }}} \right) = \frac{{2\cos i_{2} }}{{v_{1} }}Z_{1} \sin \left( {\theta_{2} - \theta_{1} } \right)\left( {\frac{{\sin \varphi_{2,1} }}{{\cos \varphi_{2,1} }} + \frac{{\sin \gamma_{2,1} }}{{\cos \gamma_{2,1} }}} \right), $$

$$ \frac{{\left( {\cos \left( {\varphi_{2,1} + \gamma_{2,1} } \right) + 1} \right)}}{{v_{\text{o}} }}\left( {\cos \gamma_{2,1} - \cos \varphi_{2,1} } \right) = \frac{{2\cos i_{2} }}{{v_{1} }}\sin \left( {\theta_{2} - \theta_{1} } \right)\sin \left( {\varphi_{2,1} + \gamma_{2,1} } \right), $$

$$ \frac{{\sin \left( {\varphi_{2,1} + \gamma_{2,1} } \right)\left( {\sin \varphi_{2,1} - \sin \gamma_{2,1} } \right)}}{{v_{\text{o}} }} = \frac{{2\cos i_{2} }}{{v_{1} }}\sin \left( {\theta_{2} - \theta_{1} } \right)\sin \left( {\varphi_{2,1} + \gamma_{2,1} } \right), $$

$$ \frac{{\left( {\sin \varphi_{2,1} - \sin \gamma_{2,1} } \right)}}{{v_{\text{o}} }} = \frac{{2\cos i_{2} }}{{v_{1} }}\sin \left( {\theta_{2} - \theta_{1} } \right), $$

$$ \frac{{\sin \varphi_{2,1} }}{{v_{\text{o}} }} - \frac{{\sin \gamma_{2,1} }}{{v_{\text{o}} }} = \frac{{2\cos i_{2} }}{{v_{1} }}\sin \left( {\theta_{2} - \theta_{1} } \right). $$

(B-2)

Using Snell’s law

$$ \frac{{\sin \varphi_{2,1} }}{{v_{\text{o}} }} = \frac{{\sin \left( {i_{2} + \theta_{2} - \theta_{1} } \right)}}{{v_{1} }}, $$

(B-3)

$$ \frac{{\sin \gamma_{2,1} }}{{v_{\text{o}} }} = \frac{{\sin \left( {i_{2} - \theta_{2} + \theta_{1} } \right)}}{{v_{1} }}, $$

(B-4)

$$ \frac{{\sin \varphi_{2,1} }}{{v_{\text{o}} }} - \frac{{\sin \gamma_{2,1} }}{{v_{\text{o}} }} = \frac{{\sin \left( {i_{2} + \theta_{2} - \theta_{1} } \right)}}{{v_{1} }} - \frac{{\sin \left( {i_{2} - \theta_{2} + \theta_{1} } \right)}}{{v_{1} }}, $$

(B-5)

inserting (B-5) into (B-2) we find

$$ \frac{{\sin \left( {i_{2} + \theta_{2} - \theta_{1} } \right)}}{{v_{1} }} - \frac{{\sin \left( {i_{2} - \theta_{2} + \theta_{1} } \right)}}{{v_{1} }} = \frac{{2\cos i_{2} }}{{v_{1} }}\sin \left( {\theta_{2} - \theta_{1} } \right), $$

$$ \frac{1}{{v_{1} }}\left[ {\sin \left( {i_{2} + \theta_{2} - \theta_{1} } \right) - \sin \left( {i_{2} - \theta_{2} + \theta_{1} } \right)} \right] = \frac{{2\cos i_{2} }}{{v_{1} }}\sin \left( {\theta_{2} - \theta_{1} } \right), $$

(B-6)

and after simplification (B-6) becomes

$$ 2\cos i_{2} \sin \left( {\theta_{2} - \theta_{1} } \right) = 2\cos i_{2} \sin \left( {\theta_{2} - \theta_{1} } \right). $$

(B-7)

The result represented by (B-7) shows that the equality given in formula (B-1) is valid.