# Shallow Velocity Estimation by Multiples for Monochannel Boomer Surveys
## 2 Travel-Time Inversion Algorithm
### Intrabed Multiples  
Another common type of multiple reflection in marine surveys is an internal bounce in the shallowest layer below the seafloor, often composed of unconsolidated sediments overlying stiffer rocks. Figure 2 shows this intrabed multiple with the ray path AA\({}^{\prime\prime}\)B\({}^{\prime\prime}\)M\({}^{\prime}\)C\({}^{\prime\prime}\)M\({}^{\prime\prime}\)C (blue dotted line). It is convenient to draw also the mirrored point B* of the bouncing point M\({}^{\prime}\), with respect to the reflection point B. In other words, we add a mirrored layer (in red) below the investigated layer.  
To estimate \(Lt\) and _Vlayer_ in this case, we proceed in a similar way as for the peg-leg multiples. The _Tabc_ travel time provides the same equation as Equation (2), and the incidence angle \(\beta\) provides the same Equation (4). The _Tib_ travel time for the intrabed multiple is slightly different, defines as follows:  
\[\textit{Tib}=2\;(\text{Taa}^{\prime\prime}+\text{Ta}^{\prime\prime}\text{b}^{ \prime\prime}+\text{Tb}^{\prime\prime}\text{m}^{\prime})=2\;(\textit{Wd}/ \textit{Vav}+2\;Lt/\textit{Vlayer})/\text{cos}\alpha \tag{22}\]  
Additionally, the fourth equation needed to complete the system is also slightly different, which is expressed as follows:  
\[\text{tan}\alpha=\text{AM}/(\text{MM}^{\prime}+\text{M}^{\prime}\text{B}+ \text{BB}^{\prime})=\text{AM}/(\text{MM}^{\prime}+2\;\text{M}^{\prime}\text{B })=\textit{X}/[2\;(\textit{Wd}+2\;Lt)] \tag{23}\]  
Figure 2: Ray paths from a source (point A) to a receiver (point C): primary reflections from the seafloor (dashed green line) and the layer bottom (continuous black line), and an intrabed multiple reflection (dashed blue line). The red lines indicate a virtual specular layer below the actual one and a virtual ray path.  
By the same process of substituting Equation (7) and rearranging, we obtain another set of nonlinear equations as follows:  
\[\textbf{{Tabc}}\cos\beta=2\left(\textbf{{Wd}}/\textbf{{V}}\textbf{{w}}+LV\right) \tag{24}\]  
\[\textbf{{Tib}}\cos\alpha=2\left(\textbf{{Wd}}/\textbf{{V}}\textbf{{w}}+2\,LV\right) \tag{25}\]  
\[\tan\beta=\textbf{{X}}/[2\left(\textbf{{Wd}}+Lt\right)] \tag{26}\]  
\[\tan\alpha=\textbf{{X}}/[2\left(\textbf{{Wd}}+2\,Lt\right)] \tag{27}\]  
Rearranging (24), we attain for \(2\,LV\)  
\[\textbf{{Tabc}}\cos\beta-2\,\textbf{{Wd}}/\textbf{{V}}\textbf{{w}}=2\,LV \tag{28}\]  
Substituting (28) in (25), and manipulating (26) and (27), we obtain  
\[\textbf{{Tib}}\cos\alpha=2\left(\textbf{{Tabc}}\cos\beta-\textbf{{Wd}}/\textbf{ {V}}\textbf{{w}}\right) \tag{29}\]  
\[Lt=0.5\,\textbf{{X}}/\tan\beta-\textbf{{Wd}} \tag{30}\]  
\[2\,Lt=0.5\,\textbf{{X}}/\tan\alpha-\textbf{{Wd}} \tag{31}\]  
Substituting (30) in (31) and rearranging, we achieve a system of two equations with two unknowns (\(\alpha\) and \(\beta\)) as follows:  
\[\textbf{{Tib}}\cos\alpha-2\,\textbf{{Tabc}}\,\cos\beta=-2\,\textbf{{Wd}}/ \textbf{{V}}\textbf{{w}} \tag{32}\]  
\[0.5\,\textbf{{X}}/\tan\alpha-\textbf{{X}}/\tan\beta=-\textbf{{Wd}} \tag{33}\]  
which can be solved numerically as in the previous case, with the same constraints (6). The layer thickness \(Lt\) can be obtained from (30), the ratio \(LV\) from (28), and the velocity _Vlayer_ from (21).  
\[LV=0.5\,\textbf{{Tabc}}\,\cos\beta-\textbf{{Wd}}/\textbf{{V}}\textbf{{w}} \tag{34}\]