Cascading image-construction operations

In order to compute the position of the image after a cascade of several image-construction operations, we first need to define the cascade sequence c as the ordered sequence of interface numbers at which we will consider that a reflection occurs. We define the source as S(-h,0) and the receiver as G(0,h), as shown in Figure . Because both S and G are at the surface, any sort of multiple event will be reflected more than once by the same interface. Therefore, the mapping of the counting index i of the cascading sequence onto the values c_i of the cascading sequence is therefore surjective, but not injective. To be able to work with indices in an efficient manner, we describe the geometry of the problem through the sequences  

$$\phi_i=\theta_{c_i}$$ (5)

and

 

l_i=d_ci

which incorporate information both on the geometry of the interfaces and on the order of the cascade, and for which the index numbering starts with the value 1. The subscripts for q will also denote the counting index for the image reflection cascade. The first reflection operation can be written as  

$${\bf q}_1={\bf A}\left(2\phi_i\right)p+{\bf b}\left(\phi_i\right).$$ (7)

Then,

$$\begin{eqnarray} {\bf q}_2&=&{\bf A}\left(2\phi_j\right){\bf q}_1+{\bf b}\left(... ...phi_j\right){\bf b}\left(\phi_i\right)+{\bf b}\left(\phi_j\right),\end{eqnarray}$$ (8) (9)

$$\begin{eqnarray} {\bf q}_3&=&{\bf A}\left(2\phi_k\right){\bf q}_2+{\bf b}\left(... ...phi_k\right){\bf b}\left(\phi_j\right)+{\bf b}\left(\phi_k\right),\end{eqnarray}$$ (10) (11)

and so on. Let us denote the counterclockwise rotation matrix with  

$${\bf R}\left(\alpha\right) = \left[ {\begin{array} {*{20}c} ... ...} \\ {\sin \alpha } & { \cos \alpha } \\ \end{array}} \right].$$ (12)

Both ${\bf A}$ and ${\bf R}$ are involutory matrices. It can be easily verified that:  

$${\bf A} \left( \alpha \right) {\bf A} \left( \beta \right) = {\bf R} \left( \alpha - \beta \right)$$ (13)

 

$${\bf R} \left( \alpha \right) {\bf R} \left( \beta \right) = {\bf R} \left( \alpha + \beta \right)$$ (14)

 

$${\bf A} \left( \alpha \right) {\bf R} \left( \beta \right) = {\bf A} \left( \alpha - \beta \right)$$ (15)

 

$${\bf R} \left( \alpha \right) {\bf A} \left( \beta \right) = {\bf A} \left( \alpha + \beta \right)$$ (16)

Chains of ${\bf A}$ operators can be written as a single operator:  

$${\bf A} \left( \beta \right) {\bf A} \left( \gamma \right) {... ...delta \right) = {\bf A} \left( \beta - \gamma + \delta \right),$$ (17)

 

$${\bf A} \left( \alpha \right) {\bf A} \left( \beta \right) {... ...ght) = {\bf R} \left( \alpha - \beta + \gamma - \delta \right),$$ (18)

According to (15), we can write any ${\bf A}$ as  

$${\bf A} \left( \alpha \right) = \left[ {\begin{array} {*{20}... ...& {-1} \\ \end{array}} \right] {\bf R} \left( - \alpha \right),$$ (19)

so the product of any number k of ${\bf A}$ operators can be written as  

$$\prod\limits_{i = 1}^k {{\bf A}\left( {\alpha _i } \right)} ... ...{i = 1}^k {\left( { - 1} \right)^{i+k-1} \alpha _i } } \right).$$ (20)

To use these properties for constructing cascades of image reflections, we must replace the set $\alpha_i$ with the inverse succession of the dips of the reflecting interfaces, multiplied by two according to the definition of ${\bf A}$ in (4):  

$$\alpha_i = 2 \phi_{k-i+1},$$ (21)

where $i=1\ldots k$. The reverse ``chronological'' order is a consequence of the operators in the chain being matrices that multiply the previous image coordinate vector from the left, as exemplified by (8) and (10). The result of the succession of image-building operations can be written as  

$${\bf q}_n = \sum\limits_{j=0}^n \left[ {\begin{array} {*{20}... ...j-1} \phi_{j-i+1} } } \right) {\bf b}\left( \phi_{n-j} \right),$$ (22)

where we define a nonphysical quantity $\phi_0=-\frac{\pi}{2}$ and we also define ${\bf b}\left( \phi_0\right)$ as the coordinates vector of the initial point in the cascade of reflections. We also consider that the summation index increases in increments of 1 and that summation operators return zero when the upper summation limit is smaller than the lower summation limit. Under the assumption that the starting point of the cascade is at the surface, and by denoting half of its x coordinate with l₀, we can write all ${\bf b}$ vectors using rotations:  

$${\bf b}\left( \phi_i\right) = 2 l_i \left[ {\begin{array} {*... ...[ {\begin{array} {*{20}c} {1} \\ {0} \\ \end{array}} \right].$$ (23)

Substituting this into (22),  

$${\bf q}_n = 2 \sum\limits_{j=0}^n l_{n-j}\left[ {\begin{arra... ...[ {\begin{array} {*{20}c} {1} \\ {0} \\ \end{array}} \right],$$ (24)

These particular choices of l₀ and $\phi_0$, together with the assumption of a surface starting point, ensure that (23) is consistent for the starting point of the cascading operations too. After a few algebraic manipulations, we obtain  

$${\bf q}_n = 2 \sum\limits_{j=0}^n l_{n-j}\left[ {\begin{arra... ..._j} \\ {\left(-1\right)^j \cos\beta_j} \\ \end{array}} \right],$$ (25)

where  

$$\beta_j = \phi_{n-j} + 2{\sum\limits_{i = 1}^j {\left( { - 1} \right)^{i+j-1} \phi_{j-i+1} } }.$$ (26)