# Shallow Velocity Estimation by Multiples for Monochannel Boomer Surveys
## 3 Synthetic Data Example  
To validate the algorithm presented above, we built a simple 1.5-D model composed of horizontal interfaces. The upper layer represents the seawater, with a constant velocity of 1500 m/s and a thickness of 20 m. In the layer below the seafloor, the velocity varied smoothly, with a Gaussian curve between 2000 and 2500 m/s (Figure 3a). Below the layer, a half-space with a velocity of 2600 m/s was placed for plotting purposes only. We simulated the travel times for 50 traces, calculating the direct arrivals, the primary reflections from the seafloor and the layer bottom, and the intrabed multiple inside that second layer. The offset was set to 2.5 m only, although values up to 10 m may be encountered in practice. We neglected the ray bending, which is fine for the water bottom primary and the direct arrival, as it is a fair approximation for the other events only when the offset between source and receiver is small with respect to the depth of the sea and the deeper interfaces. When plotting the model in the time domain (Figure 3b), the travel times of the primary reflections below the velocity anomaly showed a typical pull-up effect, distorting the actual flatness of the layer bottom. Thus, correct imaging requires quantifying and compensating that velocity anomaly. The inversion of the simulated travel times for direct arrivals, primaries, and multiples provided estimates that are astonishingly precise, with errors lower than 0.2% [17], but this encouraging result was obtained without errors or noise affecting our input data. Additionally, the forward and inverse modeling were based on the same ray tracing algorithm, so no further drawbacks appeared due to the propagation effects, such as anelastic absorption, source, and receiver radiation pattern, and incidence-angle-dependent phase rotations. For these reasons, we tested the estimation robustness with respect to data errors. The circles in Figure 4a show the estimated velocity-thickness couples in the layer below the seafloor, when perturbing the offset of each trace 40 times by random errors up to 1% while keeping the travel times of the reflections correct as modeled. Dots with the same color correspond to the same trace simulated in the model in Figure 3, i.e., in the same location for source and receiver: On the left, the traces are plotted at the model sides, where the velocity is 2000 m/s, whereas moving to the right, the traces are observed in the central part of the model, where the velocity anomaly tops to 2500 m/s. We clearly notice a linear trend for each group of perturbations for the same trace. By averaging the couples for each trace perturbation (thick blue line), the correct value of 15 m is fairly approximated for the layer thickness, while the velocity spans its actual interval. Figure 4a shows that a 1% error in the offset (i.e., 2.5 cm in this example) may produce errors up to 0.4 m (2.6%) in the layer thickness and 70 m/s (2.8%) in the layer velocity; thus, high precision is needed in the offset measurement. Effects of cable feathering and sea waves [18] must be compensated for at each trace; in addition, averaging the estimates in adjacent traces may also reduce the instabilities. The numerical solver we adopted for the equation system (24-27) was an exhaustive search in a grid 3000 \(\times\) 3000, subject to the constraints in Equation (6). More accurate (but more complex) algorithms exist, which may improve the estimation accuracy further. In actual data processing, the most challenging task is to select the multiple reflections properly. Recognizing them is just the first step. For the peg-leg multiples, we can compare the travel time from the seafloor with the time differences between other major events. Intrabed multiples are far more elusive, but crossing events may offer hints to distinguish them. However, crossing events have also a drawback--they distort the waveform and may delay or anticipate their main peaks.