# Shallow Velocity Estimation by Multiples for Monochannel Boomer Surveys
## 3 Synthetic Data Example
Another problem is the anelastic absorption, which reduces the high-frequency content in the multiples with respect to the corresponding primary, again introducing a delay in its waveform peak. Finally, assigning a precise travel time to a minimum-phase signal, as in a Boomer survey, is particularly difficult, as the waveform onset is mostly hidden by the noise or other interfering signals. For these reasons, we also tested the effect of selecting errors in the intrabed multiples and found that this is the most critical value for a reliable inversion of the layer velocity. When testing random errors in travel times up to 1%, as performed for the offsets, the results were totally disappointing. Only by reducing the perturbation percentage to 0.1%, we obtained the results in Figure 4b. The estimated thickness was slightly bumpy, but the error of the average values (blue dots) still did not exceed 1 m with respect to the actual one of 15 m. Instead, the velocity errors exceeded even 700 m/s, i.e., about 22%, the maximum value of 2500 m/s. As the travel times of the intrabed multiples were about 56 ms, the error should not exceed 56 microseconds, which was close to the sampling interval for Boomer surveys. This extreme precision is hardly achievable in a real experiment unless the data processing before selection can reduce the noise significantly and compensate for the waveforms' distortion due to the anelastic absorption. Figure 3: Synthetic model with horizontal interfaces and a Gaussian velocity anomaly in the depth domain (**a**) and in the time domain (**b**). Figure 4c,d show how the uncertainty can be reduced to achieve a better estimate for the layer velocity. The offset was then increased in both cases to 10 m, which is still manageable by a small ship in busy ports. When the perturbation was very small, i.e., 0.01% as for Figure 4b, the maximum velocity error was about 60 m/s, i.e., quite small. When we consider more realistic travel-time errors, such as 0.1% (about 560 microseconds, or 10 sample intervals), the maximum velocity error in the central anomaly was found to be 400 m/s, which is only 16%. We plotted all parts in Figure 4 using the same scale to highlight the different sensitivity of the two parameters being estimated (layer velocity and thickness) with respect to measurements that are more challenging when dealing with experimental data, i.e., offset and multiple's travel time. Minor errors in the offset (Figure 4a) caused acceptable errors in both estimates, even when using a short offset of 2.5 m. Instead, travel-time errors were much less forgiving (Figure 4b-d): They required increasing the offset to at least 10 m to achieve a result that is not ideal but still useful in practice (Figure 4d). These figures show also that, for small perturbations, the averages of estimated thickness and velocity at each location (blue lines) may be more or less bumpy but relatively close to the correct values. Thus, multiple folds of the same point in the survey can contribute significantly to the result reliability. Figure 4: The circles present the estimated layer velocity and thickness obtained by perturbing randomly the input data in different ways. Dots with the same color correspond to the same perturbed offset along the profile over the model in Figure 3. The correct thickness is 15 m, while the velocity ranges from 2000 to 2500 m/s. The thick blue line joins the averages of the perturbed offsets in each line. The perturbations are (**a**) the offset, up to 1%; (**b**) the intrabed multiple, up to 0.01% with a 2.5 m offset, (**c**) with a 10 m offset, and (**d**) with 0.1% perturbation but still with a 10 m offset.