# Shallow Velocity Estimation by Multiples for Monochannel Boomer Surveys
## 5 Discussion  
Figure 4 exhibits a linear trend of the estimates when perturbating either the offset or the multiples' travel time at a given location, highlighted by the same color of the corresponding circles. This occurred because of the small incidence angles \(\alpha\) and \(\beta\) (Figures 1 and 2) when the offset was short with respect to the layer-bottom depth. In our synthetic example, for a 2.5 m offset,  
\[\beta=\text{atan}(0.5\;\mathbf{X}/(\mathbf{W}d+\mathbf{L}t))=\text{atan}(0.5\;\times\;2.5/(20 +15))=0.035699 \tag{35}\]  
is almost equal to \(\text{tan}\beta=0.035714\). The angle \(\alpha\) is even smaller than \(\beta\), as per Equation (6). Both equation systems (8-11) and (24-27) become more linear if we approximate  
\[\text{tan}\approx\alpha,\,\text{tan}\beta\approx\beta \tag{36}\]  
using a first-order Taylor series expansion. Only the cosine terms in (8)-(9) and (24)-(25) maintain a mild nonlinearity in the equations. Thus, if perturbing one of the four variables  
Figure 7: Estimation results: (**a**) layer thickness (blue line), water depth (orange line), and offset (grey line); (**b**) layer velocity after a 5-term median filter.  
at a time, as in our examples, we may expect (quasi)linear variations in the solutions, i.e., in terms of the layer thickness and its velocity.  
The perturbation results in Figure 4 were obtained for a short offset of 2.5 m to match our field experiment and, in general, a convenient option for operations in busy or environmentally fragile areas. In the framework of crustal studies, [PERSON] [25] studied analytically the relation between offset and the uncertainties in the layer velocity and thickness. Perturbing the travel times similarly to our study, he showed that, for short offsets, variance in the estimated thickness is larger than that of velocities, while the opposite occurs in large offsets. Thus, in our case, we may rely on estimates of thickness more than velocity. When the latter parameter is more critical for offshore engineers, extending the cable may help. Of course, when reliability and resolution must be maximized, a multichannel survey and a reflection-refraction tomography provide high-quality images in depth [9], even for time-lapse surveys in complex geological structures [13; 26]. The joint inversion of direct, reflected, and refracted arrivals increases the areal coverage and local density of rays, stabilizing the tomographic inversion and increasing its robustness with respect to the noise. Therefore, when just a single channel is available, the joint inversion of different multiples is a way forward for possible improvements.  
Recognizing multiples is easy only for the \"simple\" ones, i.e., those whose travel time is (about) twice that of the corresponding primaries. In that case, the related signal amplitude is also relatively prominent. Peg-leg and intrabed multiples are weaker and more challenging, requiring ideally a preliminary tentative selection and a careful comparison of travel times: to the travel time of a deep primary, we must add the travel time of the seafloor or the estimated transit time across some layer interval. Often, this can be performed safely only in limited parts of a survey.  
The presented equations and method assume horizontal interfaces layers and homogeneous layers. When minor deviations from these hypotheses occur, the result is acceptable. Otherwise, a full tomographic approach and multichannel surveys are needed.