**3. Direct Migration of Reflections of 3D Waves from Passive Source Based on Geometry**

*3.1. Principle of Calculation of Direct Migration*

The principle of calculating random noise by direct migration can be expressed as follows:

$$I(\mathbf{x}) = \sum\_{k=1}^{N\_{\mathbf{v}}} \int\_{0}^{T} R\_{K}^{A}(\mathbf{x}, t) R\_{K}^{B}(\mathbf{x}, t) dt \tag{2}$$

*R<sup>A</sup> <sup>K</sup>* (*x*, *<sup>t</sup>*) and *<sup>R</sup><sup>B</sup> <sup>K</sup>*(*x*, *t*) are the wave fields from the *k*th (*k* = 1, 2, ... , *Ns*) source at a certain time, and are received by the receivers and on the surface. The two are extended downward, and the underground image of the reflection *I*(*x*) can be obtained through correlation (Figure 16). The final result of imaging is the sum of the results of correlation of a series of sources and receivers in a series of time windows. Artman [27] summarized the process of calculation of direct migration of data from a passive source as the extrapolation, correlation, and summation of the wave field. All algorithms for migration imaging that involve the extrapolation of the wave field and are conducive for use in the relevant imaging conditions can apply the direct migration method

**Figure 16.** (**a**) Receivers A and B receive a single vibration from underground at a certain time. (**b**) The image of the underground reflection is obtained by downward continuation. (**c**) Diagram of correlation.

We used the one-way wave pre-stack depth migration method to process the original random noise records. It consisted of two steps: recursion of the source and receiver points along the direction of the depth, and imaging [33] to satisfy the requirements of the direct migration method.

The forward calculation of the 2D passive source by using direct migration was carried out by using the models of velocity and random noise records described in Section 2, and shown in Figures 2 and 3, to illustrate the process of imaging of the reflections by the passive source. One of the random noise records was set as the source point, with the remaining records used for one-way wave migration. All 301 data sources were calculated and stacked, and the migration profile was obtained by using conventional seismic processing, as shown in Figure 17.

One channel of the random noise records was set as the source point, and the other 300 channels were set as receiver points, that is, the range of geometry included all of the data. Performing one-way wave migration and stacking for all 301 "sources" and models of velocity yielded a direct migration profile (Figure 17b). The results were consistent with those of migration imaging based on the conventional processing of the 2D passive source in Figure 17a. That is, the direct migration imaging of reflections by a passive source was accurate, and the proposed method is thus feasible.

**Figure 17.** Comparison of (**a**) results of migration imaging of a 2D passive source using conventional processing and (**b**) direct migration imaging.
