# Shallow Velocity Estimation by Multiples for Monochannel Boomer Surveys
## 2 Travel-Time Inversion Algorithm
### Peg-Leg Multiples  
The _Tabc_ and _Tpl_ travel times provide the following two equations:  
\[\textbf{{Tabc}}=\text{Taa}^{\prime}+\text{Ta}^{\prime}\textbf{b}+\text{Tbb}^{ \prime}+\text{Tb}^{\prime}\text{c}=2\text{ (Taa}^{\prime}+\text{Ta}^{\prime}\textbf{b})=2\text{ (}\textbf{{Wd}}/\textbf{{Vw}}+\textit{{Lt}}/\textit{{Vlayer}})/\text{cos}\beta \tag{2}\]  
Figure 1: 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 a peg-leg multiple reflection (dashed blue line).  
\[Tpl=2\;(\text{Taa}^{\prime\prime}+\text{T}^{\prime\prime}\text{b}^{\prime\prime}+ \text{T}\text{m}^{\prime}\text{m}^{\prime\prime})=2\;(2\;\mathbf{Wd}/\mathbf{Vv}+Lt/\text{Vlayer})/ \text{cos}\alpha \tag{3}\]  
where two additional unknowns \(\alpha\) and \(\beta\) appear (Figure 1). They are the incidence angles for the peg-leg multiple and for the primary reflection from the lower layer bottom. We need two more equations to obtain a viable solution, which can be obtained by trigonometric relations as follows:  
\[\tan\beta=\text{AM}/(\text{MM}^{*}+\text{M}^{*}\text{B})=\mathbf{X}/[2\;(\mathbf{Wd}+Lt )] \tag{4}\]  
\[\mathbf{X}=2\;[(\text{MM}^{*}+\text{M}^{*}\text{B})+\text{MM}^{*}]\;\text{tan} \alpha=2\;(2\;\mathbf{Wd}+Lt)\;\text{tan}\alpha \tag{5}\]  
Equations (2)-(5) form a nonlinear system in the four unknowns \(\alpha\), \(\beta\), \(Lt\), _Vlayer_ that can be solved numerically with a few constraints as follows:  
\[0<\alpha<\beta<\gamma=\text{atan}(0.5\;\mathbf{X}/\mathbf{Wd}),\;\mathbf{X}>0,\;Lt>0 \tag{6}\]  
where \(\gamma\) is the incidence angle of the primary reflection from the seafloor. To reduce the computational efforts, we can simplify this equation system by defining the ratio \(LV\) as  
\[LV=Lt/\text{Vlayer} \tag{7}\]  
Substituting (7) in (2) and (3) and rearranging, we obtain  
\[\mathbf{Tabc}\;\text{cos}\beta=2\;(\mathbf{Wd}/\mathbf{Vv}+LV)=2\;\mathbf{Wd}/\mathbf{Vv}+2\;LV \tag{8}\]  
\[\mathbf{Tpl}\;\text{cos}\alpha=2\;(2\;\mathbf{Wd}/\mathbf{Vv}+LV)=4\;\mathbf{Wd}/\mathbf{Vv}+2\;LV \tag{9}\]  
\[\text{tan}\beta=\mathbf{X}/[2\;(\mathbf{Wd}+Lt)] \tag{10}\]  
\[\text{tan}\alpha=\mathbf{X}/[2\;(2\;\mathbf{Wd}+Lt)] \tag{11}\]  
By subtracting (8) from (9), we obtain a system of three equations with three unknowns (\(\alpha\), \(\beta\), and \(Lt\)) as follows:  
\[\mathbf{Tpl}\;\text{cos}\alpha-\mathbf{Tabc}\;\text{cos}\beta=2\;\mathbf{Wd}/\mathbf{Vv} \tag{12}\]  
\[\text{tan}\beta=\mathbf{X}/[2\;(\mathbf{Wd}+Lt)] \tag{13}\]  
\[\text{tan}\alpha=\mathbf{X}/[2\;(2\;\mathbf{Wd}+Lt)] \tag{14}\]  
Rearranging (13) and (14), we obtain  
\[2\;\mathbf{Wd}+2\;Lt=\mathbf{X}/\text{tan}\beta \tag{15}\]  
\[4\;\mathbf{Wd}+2\;Lt=\mathbf{X}/\text{tan}\alpha \tag{16}\]  
Subtracting (12) from (13), together with (12), we obtain a system of two nonlinear equations with two unknowns (\(\alpha\) and \(\beta\)) as follows:  
\[\mathbf{Tpl}\;\text{cos}\alpha-\mathbf{Tabc}\;\text{cos}\beta=2\;\mathbf{Wd}/\mathbf{Vv} \tag{17}\]  
\[\mathbf{X}/\text{tan}\alpha-\mathbf{X}/\text{tan}\beta=2\;\mathbf{Wd} \tag{18}\]  
which can be solved numerically with the constraints (5). When \(\alpha\) and \(\beta\) are computed, we can obtain the layer thickness \(Lt\) using Equation (13) or (14). For example, from (14),  
\[Lt=0.5\;\mathbf{X}/\text{tan}\alpha-2\;\mathbf{Wd} \tag{19}\]  
where we use the bold font for \(\mathbf{tan}\alpha\) to highlight that the \(\alpha\) angle becomes a known value at this point. The \(LV\) product, similarly, can be obtained by either (8) or (9). Using (8), for example, we have  
\[LV=0.5\;\mathbf{Tabc}\;\text{cos}\beta-\mathbf{Wd}/\mathbf{Vv} \tag{20}\]Substituting \(L\)\(V\) in (7), we obtain the last unknown _Vlayer_ as follows:  
\[\textit{Vlayer}=\textit{L}t/\textit{L}V \tag{21}\]