**4. Discussion**

In this section, the estimation errors of two-way reflected time and velocity are discussed. Take the first slice in Figure 8b for instance, a 3D surface plot can be computed, which is shown in Figure 15a. The maximum value in this figure indicates the two-way reflected time and velocity of this hyperbola. However, some error will generate from the noise interference and incomplete hyperbola; the correct point is probably in a region; a flat at an amplitude of 0.8 is selected to split out this region. Two profiles, which are the red rectangle and black rectangle in Figure 15, are extracted out to compute the error intervals of velocity and reflected time, respectively. The two intersections of the blue curve and gray line around the maximum value are selected to compose the error limit.

**Figure 15.** Error computation. (**a**) 3D surface plot of the first slice in Figure 8b. (**b**) The profile of in the red rectangle of (**a**). (**c**) The profile of in the black rectangle of (**a**).

In Figure 15, the error ranges of velocity and two-way reflected time are 0.161–0.179 m/ns and 55.8–56.84 ns, respectively. So, the real velocity and time may be within these two ranges. A maximum relative error (MRE) can be adopted to quantitatively describe the error:

$$\begin{cases} MRE\_v = \left| \max(v - v\_{ran}) \right| / v \\ MRE\_t = \left| \max(t - t\_{ran}) \right| / t \end{cases} \tag{11}$$

where *v* and *t* are the estimated values using 3D velocity spectrum, *vran* and *tran* represent the error ranges. So, the *MREv* and *MREt* in Figure 13 are 5.29% and 1.05%, respectively.

Subsequently, the errors of all hyperbolas are computed. The average MREs of time and velocity are 0.68% and 7.99%, respectively. It is clear that the error of two-way reflected time can be omitted, which is far less than the error of velocity. Furthermore, we computed the distributions of velocities and error ranges in horizontal and vertical directions which are shown in Figure 16.

**Figure 16.** Distributions of velocities and error ranges. (**a**) Velocities and error ranges with distance. (**b**) Velocities and error ranges with depth. The triangular points represent the error ranges; the blue dots represent the estimated velocities.
