**1. Introduction**

Passive-source seismic exploration, which is based on seismic wave interferometry, was developed by Schuster [1] and Bakulin and Calvert [2]. This method is different from active-source seismic exploration in that it does not require excitation by an artificial source or knowledge of the number and locations of the sources. Correlations can be used to convert environmental seismic noise or micro-earthquakes into a determined seismic response. The theory of seismic interferometry involves "perform[ing] correlations on the seismic records received by receivers at two points on the surface, and the results can be regarded as the records of the wave field of the active source, with one point as the receiver and the other point as the source" [3,4]. Applying this calculation to all random signals of the receiver can yield a group of virtual shot records of the passive source that are similar to forward records of the active source in seismic exploration.

Methods of passive-source seismic exploration can be divided into techniques for investigating passive-source surface waves [5–7] and passive-source body waves [3,8–11], according to the types of seismic waves. Because the Earth absorbs body waves to a much greater extent than surface waves, it is much more difficult to capture images of the former than the latter [12]. Some researchers have retrieved reflections of waves from ambient noise by using illumination-based diagnosis [13,14], while others have sought to improve the quality of data on reflections of the passive source [15–19].

Methods of exploring reflections of the passive source have been applied in many areas. Masatoshi et al. [20] calculated the velocities of the P-wave and the S-wave, the

**Citation:** Liu, Y.; Liu, G. Three-Dimensional Processing of Reflections for Passive-Source Seismology Based on Geometric Design. *Appl. Sci.* **2023**, *13*, 6126. https://doi.org/10.3390/app13106126

Academic Editor: José A. Peláez

Received: 18 March 2023 Revised: 15 May 2023 Accepted: 15 May 2023 Published: 17 May 2023

**Copyright:** © 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

bundle factor of the P-wave, and other structural parameters based on seismic noise in field data obtained from observation wells in Cold Lake, Alberta, Canada. Seismic wave interference technology was used to investigate the presence of metal ores by Cheraghi et al. [21]. Cheraghi et al. [22] applied passive source exploration to the four-dimensional dynamic monitoring of carbon dioxide storage sites at an Aquistore facility. Eric et al. [23] used data on underground activity in a mine and vibrations of the ventilation pipes in the context of exploring mining resources, and Brenguier et al. [24] used the noise in waves generated by a railway to monitor the shallow crust. Brenguier et al. [25] also used a receiver array to monitor long-term seismic velocity and anticipate geological events.

According to the theory of passive seismic exploration, sources of random noise are needed to evenly identify geological targets from all directions and obtain accurate kinematical results. Artifacts may be obtained when the source is not located along the direction of the receiver line [14]. Three-dimensional (3D) passive-source seismic exploration can be used to obtain more uniform information and images of better quality than twodimensional (2D) passive seismic exploration. Draganov et al. [12] and Nakata et al. [26] obtained the underground velocity from environmental noise by using seismic wave interference technology and generated a corresponding 3D seismic profile. Chamarczuk et al. [27] used the full-scale 3D seismic method of the virtual source survey to explore the Kylylahti polymetallic mine in Finland and obtained geological information that could not be obtained through active seismic exploration.

While 3D passive seismic reflections have delivered promising results, they require intensive computations as well as a large storage space for the data obtained. Given these issues, we propose a method to calculate images of 3D reflections based on the geometric design of the area. We use information on the area, the surface, and the underground structure to design a reasonable geometry for it, subsequently use that to screen for effective data to obtain correlations, and finally generate an imaging profile with uniform folds and azimuths. The passive-source direct migration method is another means to improve computational efficiency. It was proposed by Artman [28] and has been applied to passive seismic exploration [29–32]. The passive-source direct migration method does not generate correlations to obtain virtual shot records, but directly performs migration imaging to obtain records of random noise and improve computational efficiency.

### **2. Extracting 3D Virtual Shot Records Based on Geometry**

### *2.1. Principle of Reflection of Waves from Passive Sources*

Interferometry is the basis of calculations for passive seismic exploration and can be derived from the equation of the acoustic wave according to the reciprocity theorem [31,32]:

$$2R\left\{G(\mathbf{x}\_{A,\prime},\mathbf{x}\_{\mathcal{B},\prime}\omega)\right\}S(\omega) = \frac{2}{v\rho}\left\langle u(\mathbf{x}\_{A\prime}\omega)u(\mathbf{x}\_{\mathcal{B},\prime}\omega)\right\rangle\tag{1}$$

where *G*(*xA*,, *xB*,*ω*) is Green's function of the observation points *xA* and *xB* in the frequency domain, *ω* is the angular frequency, *S*(*ω*) is a random source, and *v* and *ρ* are the velocity and the density of medium, respectively. *u*(*xA*, *ω*) and *u*(*xB*, *ω*) are random signals received by the receivers, ∗ represents the superposition of the results of the mutual interferometry of different windows, and *R*{ ∗} is calculated by taking the real part.

Figure 1 shows that the above formula can be used to obtain virtual shot records similar to those due to excitation by an active source, with one point as the source and the other as the receiver, by performing interferometry on two points of the passive source. If point A does not move, and point B is transformed, we can obtain virtual shot records with point A as the source.

**Figure 1.** Schematic diagram of reflection interferometry.

We use a 2D forward model of a passive source to illustrate the process of extraction of the virtual shot records of the reflections by the passive source. We established a model of velocity as shown in Figure 2a, with a length of 3000 m, a depth of 800 m, and a grid spacing of dx = dz = 10 m. The lower, left, and right boundaries of the model were the boundaries of absorption, and we set 31 absorption layers. The upper boundary was a free boundary. The model contained three geological horizons with velocities of 1000, 2500, and 4000 m/s from top to bottom. A receiver was arranged after every 10 m on the surface, with a total of 301 receivers. There were 2000 sources of random noise (Figure 2b), and 60 s of random noise records were obtained through forward modeling (Figure 3). The source was made to continuously shake during acquisition, and its amplitudes and phases were completely random. Each of the channels of random noise was related to the other 300 channels. That is, one channel of the seismic record of a virtual source was chosen as the source, and the others were used as receivers (Figure 4a). The signal-to-noise ratios (SNRs) of the near-offset data in the red box in the figure were high. The profile of migration was obtained, as shown in Figure 4b, by calculating all 301 respective channels, using the conventional method of processing seismic data.

**Figure 2.** (**a**) Model of velocity. (**b**) Distribution of the sources of random noise.

**Figure 3.** Random noise records lasting 60 s.

**Figure 4.** (**a**) Virtual shot records (the source point x = 1500 m). (**b**) Migration profile obtained by conventional processing.

### *2.2. Calculating 3D Virtual Shot Records of Passive Source Based on Geometry*

The core of passive seismic reflection exploration is to obtain virtual shot records based on correlations between a channel of ambient noise records and other channels. When the length of the correlation is given, the ambient seismic records received over a long time need to be divided into several windows according to this length, and can then be correlated and stacked. To reconstruct a high-fidelity Green function, the time series of mutual correlation among the seismic data is usually very long. This makes it necessary to increase the number of folds to improve the quality of the Green function, which in turn reduces the computational efficiency of imaging of the passive source. Therefore, the number of virtual shot records, length of the ambient seismic records, length of the correlation window, and number of seismic channels involved in the calculation all influence the time needed to generate the virtual shot records.

Passive seismic exploration using reflections of waves from the source is based on correlations between all channels of random signals, and the generated virtual shot records are similar to those of an active source. They should also comply with the corresponding requirements of geometric design in theory. An appropriate geometry can not only improve the quality of imaging, but can also reduce the computational expense. Many seismic channels are involved in correlation and stacking in the calculation of virtual shot records, and the long time needed to record the ambient noise incurs a high computational intensity. The range of seismic channels needed to obtain correlations in 2D calculations is usually selected by setting the spacing and range of offset of points on the source. However, a rule is required in case of 3D calculations to ensure the quality of the generated virtual shot records. The data processing features uniform folds, wide azimuths, and other requirements that are similar to those for active seismic exploration. The geometric design has been extensively studied in the context of conventional seismic exploration, and exploration based on reflections from a passive source can directly learn from the experience accumulated by this research. We designed a method to compute the virtual shot records of reflections from a 3D passive source based on the design of the geometry for active-source seismic data to ensure a sufficient number of folds and quality of imaging while reducing the computational cost.

Before acquiring 3D data on the active source for seismic exploration, the geometry required to determine how to position the sources and receivers can be designed according to the known surface and the underground geological conditions of the given area. One can then specify the location of each source and receiver and the corresponding relationships between them. Following this, we must ensure that the collected data contain uniform folds, and then set the range of the offset distance and the azimuths. The geometric design usually considers the geological target elements and cost of acquisition. The file containing these geometric data is usually called the Shell Processing Support (SPS) Format for 3D Land Survey, and is a standard file for recording information on the source and the receiver as well as their relationship. The SPS file is mainly composed of a source-information file (S file, Table 1), a receiver-information file (R file, Table 2), and a source–receiver relationship file (X file, Table 3). The SPS file determines the unique physical location of the sources and receivers in the working area as well as their relationships. In the context of correlations, the SPS.S file stores information on the sources of all virtual shot records, and the SPS.X file contains the range of the receivers in the SPS.R file that need to be correlated with these sources.


**Table 1.** Key header words in the SPS.S (source) file.

**Table 2.** Key header words in the SPS.R (receiver) file.


**Table 3.** Key header words in the SPS.X (relation) file.


The difference between the geometry of the reflections from a passive source and that used in conventional seismic exploration should be considered to formulate the geometric design. Because each point representing a receiver in passive seismic exploration can become a point representing a source through correlation, the density of the points representing the source in passive seismic exploration may be much higher than that in active seismic exploration. On the premise of satisfying the requirements related to the number of folds and the SNR of the data, it is thus necessary to set an appropriate number of passive sources and receivers to significantly reduce the cost of processing.

Based on the above requirements, we propose a method to generate virtual shot records of 3D reflections from a passive source based on the above-mentioned geometry file, as shown in Figure 5. The procedure is as follows: (1) Having collected random noise records from the field, a geometry with a sufficient number of folds and uniform azimuths should be designed by using a design software according to the known surface, information on the underground structure, and geological targets, and should be stored in the SPS file. (2) The information is obtained on the source and receivers from the SPS.X file. The length of the correlation should be determined according to the conditions of the working area. After completion of the generation of virtual data for a receiver from its related shots, the initial data of this receiver are obtained for one correlation time. Then, we slide the time window by a correlation time, and repeat the above process until the end of the random noise records. By vertically stacking all of the correlation results, we can obtain the seismic record of this receiver. Following this, the remaining receivers corresponding to the source are calculated to obtain one shot of the virtual seismic records. (3) The entire SPS.X file is read to obtain all of the virtual shot records according to the previous step, and the imaging

section is then obtained by processing seismic records of reflections of waves from the passive source according to the conventional method of processing.

**Figure 5.** Correlation processing using the SPS file.
