Seismic Interferometry by Multi-Dimensional Deconvolution Free from Point-Spread Function

Abstract

Seismic interferometry (SI) can be used to reconstruct a pseudo-acquisition from response of a passive source, while the reconstructed data can be used to recover a portion of the model space that is different from that recovered by the inversion of original measurements. The SI by crosscorrelation requiring the seismic wave field to be evenly distributed, which limits the scope of application of this method. SI by multi-dimensional deconvolution (MDD) broken through the limitation that the wave field must be evenly distributed, but there are still some limitations in some special practices. Interferometric point-spread function (PSF), like the correlation function, is a necessary condition of the MDD method, but in some practices it cannot be derived from the field data. A new MDD method is proposed in this paper that free from PSF, and theoretically proved that it is equivalent to usual MDD. We demonstrate the effectiveness of this method with two numerical examples of one-sided illumination, and the source blurring phenomenon in the results of SI by crosscorrelation is effectively eliminated. The first numerical example is far-field one-side illumination, which can also be treated with the usual MDD, and the comparison of the results shows that the two MDD methods are equally effective. The second example is the near-field one-side illumination, and only the MDD method proposed in this paper can be used because the PSF cannot be obtained.

1 Introduction

Passive seismic is seismic imaging using sources of opportunity [1]. This method does not use standard air guns, vibrators or dynamite, but only deploys an array of seismic acquisition equipment to record the seismic response of opportunistic source. Sources of opportunity include microseisms, ambient-noise, industrial noise, etc. Because sources of opportunity exist for a long time relative to artificial sources, they are suitable for time-lapse seismic assessments [1]. The main industrial applications of passive seismic time-lapse assessments include: reservoir monitoring [2], stress redistribution around the coal longwall mining panel [3,4,5], geological stability of train tunnel [6]. The seismic response of passive source cannot be directly used for imaging, because the passive source often has some obvious shortcomings, such as the uncertainty of source location and of shot time, and multi-sources mixing, etc. In order to use the passive sources to imaging the target medium, their seismic responses need to be reconstructed. Reconstructing the seismic response of a passive source into the seismic response between geophone pairs is a method known as seismic interferometry (SI). This method is also called Green’s function extraction method, or virtual source method, because one of the geophones appears to be a “virtual” source for reconstructing seismic response.

SI is a rapidly developing research field in recent years, which generally reconstructs the Green’s function between two geophones by the cross-correlation method. Compared with the SI by cross-correlation, the deconvolution interferometry proposed by Snieder et al. [7] has some advantages; for example, it can compensate for the properties of the source wavelet, and there is no need to assume that the medium is lossless [8]. Therefore, this method has a good performance for recovering the impulse response from noise records by a long and complicated source-time function. Nakata et al. [9] applied the cross-coherence method to the SI of traffic noise to retrieve both body waves and surface-waves. By using only the phase information and ignoring amplitude information, the method effectively removes the source signature from the extracted response and yields a stable structural reconstruction even in the presence of strong noise. These methods belong to 1D SI.

All of the methods mentioned above require the wave field to be evenly distributed. The Green’s function between two points could be recovered using the cross-correlation function of the ambient noise measured at these two points, because the ambient-noise is more uniform from all directions [10, 11]. However, when the industrial noise is used as the source, one-sided illumination can lead to severe distortions of the retrieved Green’s function, which is proportional to a Green’s function with a blurred source. These limitations can be partially solved by MDD methods [12]. The source blurring of cross-correlation can be quantified by the point-spread function (PSF), which can be obtained from the correlation function of the data [13]. The source of the Green’s function obtained by the correlation method can be deblurred by deconvolving the correlation function for the PSF [8]. MDD greatly expands the scope of application of SI. However, for a type of thread sources in tunnels, such as the belt conveyor in the coal working face tunnels or subway trains, it is difficult to get PSF from the field data, because the geophones and the thread sources in the same tunnel. In the studies conducted by Wapenaar et al. [14], similar problems were encountered when trying to apply MDD for deblending data from simultaneous sources, where the source and receiver arrays even coincided. The results indicated that in the cases where the sources and receivers were very close together, it was not possible to invert the PSF using conventional MDD-methodology. The aforementioned researchers believed that this problem had occurred due to the fact that the spatial spectrum of the PSF was too broad. They reasoned that the problem could potentially be mitigated by extending the overburden, or possibly by including an additional spatial filter inside their formulation. Lu found that another PSF could be obtained from the data of the adjacent tunnel, and a new MDD formula based on the new PSF is derived [15].

In this paper, we propose a new SI by MDD method which free from PSF, which is suitable for practices where PSF cannot be obtained from field data. This method is similar to a multi-dimensional whitening filter, which can be proved to be partly equivalent to usual MDD method.

2 Green’s Function Representation and SI by MDD

Figure 1a shows the illustration of SI by MDD for the situation of direct-wave interferometry. In an arbitrary inhomogeneous anisotropic dissipative medium, a volume $V$ was defined, which enclosed by a surface $S$ with outward pointing normal vector ${{\mathbf{n}}} = \left( {n_{1} ,n_{2} ,n_{3} } \right)$. Surface $S$ was coincided with the surface $S_{rec}$ and a hemisphere $S_{1}$. For the practices discussed in this paper, $S_{rec}$ stands for tunnels in which geophone array ${{\mathbf{x}}}$ is arranged, and ${{\mathbf{x}}}_{B}$ is a geophone in an adjacent tunnel. A straightforward crosscorrelation of responses at ${{\mathbf{x}}}$ and ${{\mathbf{x}}}_{B}$ gives a set of impulse response (Green’s function) at one receiver ${{\mathbf{x}}}_{B}$ of virtual sources at the position of ${{\mathbf{x}}}$. The mass density in $V$ is $\rho$. The sources are outside of $V$, and the i-th source is named ${{\mathbf{x}}}_{S}^{\left( i \right)}$. We define the Fourier transform of $G\left( {{{\mathbf{x}}}_{B} ,{{\mathbf{x}}}_{S} ,t} \right)$ as

$$ \hat{G}\left( {{{\mathbf{x}}}_{B} ,{{\mathbf{x}}}_{S} ,\omega } \right) = \mathop \smallint \limits_{ - \infty }^{\infty } \exp \left( { - j\omega t} \right)G\left( {{{\mathbf{x}}}_{B} ,{{\mathbf{x}}}_{S} ,t} \right){\text{d}}t, $$

Fig. 1

a Illustration of interferometry by MDD for the situation of direct-wave interferometry. b Illustration of the point-spread function $\varGamma \left( {{{\mathbf{x}}},{{\mathbf{x}}}_{A} ,t} \right)$ (Eq. 7) (Modified from [13])

with $j$ the imaginary unit and $\omega$ the angular frequency. The convolution-type Green’s function representation is given by [13]

$$ \begin{aligned} \hat{G}\left( {{{\mathbf{x}}}_{B} ,{{\mathbf{x}}}_{S} ,\omega } \right) & = - \oint\limits_{S} {\frac{1}{{j\omega \rho \left( {{\mathbf{x}}} \right)}}\left( {\partial_{i} \widehat{{\overline{G}}}\left( {{{\mathbf{x}}}_{B} ,{ }{{\mathbf{x}}},\omega } \right)\hat{G}\left( {{ }{{\mathbf{x}}},{{\mathbf{x}}}_{S} ,\omega } \right)} \right.} \\ & \quad \left. { - \widehat{{\overline{G}}}\left( {{{\mathbf{x}}}_{B} ,{ }{{\mathbf{x}}},\omega } \right)\partial_{i} \hat{G}\left( {{ }{{\mathbf{x}}},{{\mathbf{x}}}_{S} ,\omega } \right)} \right)n_{i} {\text{d}}{{\mathbf{x}}} \\ \end{aligned} $$

(1)

(Einstein’s summation convention applies to repeated lower case Latin subscripts). The notation $\widehat{{\overline{G}}}$ is introduced to denote a reference state with possibly different boundary conditions at $S$ and/or different medium parameters outside $S$ (but in $V$ the medium parameters for $\widehat{{\overline{G}}}$ are the same as those for $\hat{G}$). The bar is usually omitted because $\hat{G}$ and $\widehat{{\overline{G}}}$ are usually defined in the same medium throughout space. $\partial_{i}$ denotes that the differentiation is carried out with respect to the components of $n_{i}$.

The convolution-type Green’s function representation (Eq. (1)) is a basic expression for SI by MDD in open systems [16]. $\hat{G}\left( {{ }{{\mathbf{x}}},{{\mathbf{x}}}_{S} ,\omega } \right)$ is writed as the superposition of an inward- and outward-propagating part at x on $S_{rec}$, according to $\hat{G}\left( {{ }{{\mathbf{x}}},{{\mathbf{x}}}_{S} ,\omega } \right) = \hat{G}^{in} \left( {{ }{{\mathbf{x}}},{{\mathbf{x}}}_{S} ,\omega } \right) + \hat{G}^{out} \left( {{ }{{\mathbf{x}}},{{\mathbf{x}}}_{S} ,\omega } \right)$. After a series of simplifications of the integral of Eq. (1), convolution-type representation is obtained [13]:

$$ G\left( {{{\mathbf{x}}}_{B} ,{{\mathbf{x}}}_{S} ,t} \right) = \mathop \smallint \limits_{{S_{rec} }}^{ } \overline{G}_{d} \left( {{{\mathbf{x}}}_{B} ,{ }{{\mathbf{x}}},t} \right) \otimes { }\hat{G}^{in} \left( {{{\mathbf{x}}},{{\mathbf{x}}}_{S} ,t} \right){\text{d}}{{\mathbf{x}}} $$

(2)

where, $\otimes$ denotes temporal convolution.

The passive source is regarded as a series of discrete sources ${{\mathbf{x}}}_{S}^{\left( i \right)}$, and the wavelet of each discrete source is $\omega^{\left( i \right)} \left( t \right)$. The convolution of Green’s function and wavelet $\omega^{\left( i \right)} \left( t \right)$ was used to obtain the corresponding wave displacement [13]:

$$ u^{in} \left( {{{\mathbf{x}}},{{\mathbf{x}}}_{S}^{\left( i \right)} ,t} \right) = G^{in} \left( {{{\mathbf{x}}},{{\mathbf{x}}}_{S}^{\left( i \right)} ,t} \right) \otimes \omega^{\left( i \right)} \left( t \right) $$

(3)

$$ u\left( {{{\mathbf{x}}}_{B} ,{{\mathbf{x}}}_{S}^{\left( i \right)} ,t} \right) = G\left( {{{\mathbf{x}}}_{B} ,{{\mathbf{x}}}_{B}^{\left( i \right)} ,t} \right) \otimes \omega^{\left( i \right)} \left( t \right) $$

(4)

Then, by substituting Eqs. (3) and (4) into Eq. (2), the following was obtained [13]:

$$ u\left( {{{\mathbf{x}}}_{B} ,{{\mathbf{x}}}_{S}^{\left( i \right)} ,t} \right) = \mathop \smallint \limits_{{S_{rec} }}^{ } \overline{G}_{d} \left( {{{\mathbf{x}}}_{B} ,{{\mathbf{x}}},t} \right) \otimes {\text{u}}^{in} \left( {{{\mathbf{x}}},{{\mathbf{x}}}_{S}^{\left( i \right)} ,t} \right){\text{d}}{{\mathbf{x}}} $$

(5)

In order to solve the above formula in the terms of a least square, the incident wave ${\text{u}}^{in} \left( {{{\mathbf{x}}}_{A} ,{{\mathbf{x}}}_{S}^{\left( i \right)} ,t} \right)$ was cross-correlated on both sides, and ${{\mathbf{x}}}_{A}$ represented one point in ${{\mathbf{x}}}$. Subsequently, the following was obtained [13]:

$$ C\left( {{{\mathbf{x}}}_{B} ,{{\mathbf{x}}}_{A} ,t} \right) = \mathop \smallint \limits_{{S_{rec} }}^{ } \overline{G}_{d} \left( {{{\mathbf{x}}}_{B} ,{ }{{\mathbf{x}}},t} \right) \otimes \varGamma \left( {{{\mathbf{x}}},{{\mathbf{x}}}_{A} ,t} \right){\text{d}}{{\mathbf{x}}} $$

(6)

where, the PSF is defined as

$$ \begin{aligned} \varGamma \left( {{{\mathbf{x}}},{{\mathbf{x}}}_{A} ,t} \right) & = \mathop \sum \limits_{i} {\text{u}}^{in} \left( {{{\mathbf{x}}},{{\mathbf{x}}}_{S}^{\left( i \right)} ,t} \right) \otimes {\text{u}}^{in} \left( {{{\mathbf{x}}}_{A} ,{{\mathbf{x}}}_{S}^{\left( i \right)} , - t} \right) \\ & = \mathop \sum \limits_{i} {\text{G}}^{in} \left( {{{\mathbf{x}}},{{\mathbf{x}}}_{S}^{\left( i \right)} ,t} \right) \otimes {\text{G}}^{in} \left( {{{\mathbf{x}}}_{A} ,{{\mathbf{x}}}_{S}^{\left( i \right)} , - t} \right) \otimes W^{\left( i \right)} \left( t \right) \\ \end{aligned} $$

(7)

and the correlation function (also known as virtual shot gather) is defined as

$$ \begin{aligned} C\left( {{{\mathbf{x}}}_{B} ,{{\mathbf{x}}}_{A} ,t} \right) & = \mathop \sum \limits_{i} u\left( {{{\mathbf{x}}}_{B} ,{{\mathbf{x}}}_{S}^{\left( i \right)} ,t} \right) \otimes {\text{u}}^{in} \left( {{{\mathbf{x}}}_{A} ,{{\mathbf{x}}}_{S}^{\left( i \right)} , - t} \right) \\ & = \mathop \sum \limits_{i} G\left( {{{\mathbf{x}}}_{B} ,{{\mathbf{x}}}_{S}^{\left( i \right)} ,t} \right) \otimes {\text{G}}^{in} \left( {{{\mathbf{x}}}_{A} ,{{\mathbf{x}}}_{S}^{\left( i \right)} , - t} \right) \otimes W^{\left( i \right)} \left( t \right) \\ \end{aligned} $$

(8)

with $W^{\left( i \right)} \left( t \right) = \omega^{\left( i \right)} \left( t \right) \otimes \omega^{\left( i \right)} \left( { - t} \right)$, and ${\Sigma }$ denotes the superposition of all the sources.

$\varGamma \left( {{{\mathbf{x}}},{{\mathbf{x}}}_{A} ,t} \right)$, the interferometry PSF, as defined in Eq. (7) is the crosscorrelation of the inward-propagating wavefields at ${{\mathbf{x}}}$ and ${{\mathbf{x}}}_{A}$, summed over the sources [17], see Fig. 1b. When the medium is lossless and the wavefield is equipartitioned, the PSF would approach a temporally and spatially band-limited delta function [13]. When these assumptions are violated, Eq. (6) shows that the PSF blurs the source of Green’s function in the spatial directions [7]. MDD involves inverting Eq. (6). This removes the distorting effects of the PSF $\varGamma \left( {{{\mathbf{x}}},{{\mathbf{x}}}_{A} ,t} \right)$ from the correlation function $C\left( {{{\mathbf{x}}}_{B} ,{{\mathbf{x}}}_{A} ,t} \right)$ and yields an improved estimate of the Green’s function $\overline{G}_{d} \left( {{{\mathbf{x}}}_{B} ,{ }{{\mathbf{x}}},t} \right)$ [13].

Then, through the discretization of the integral in Eq. (6), the following was obtained [13]:

$$ C\left( {{{\mathbf{x}}}_{B} ,{{\mathbf{x}}}_{A}^{\left( l \right)} ,t} \right) = \mathop \sum \limits_{k} \overline{G}_{d} \left( {{{\mathbf{x}}}_{B} ,{{\mathbf{x}}}^{\left( k \right)} ,t} \right) \otimes \varGamma \left( {{{\mathbf{x}}}^{\left( k \right)} ,{{\mathbf{x}}}_{A}^{\left( l \right)} ,t} \right) $$

(9)

For all ${{\mathbf{x}}}_{A}^{\left( l \right)}$ on the receiver surface $S_{rec}$.

In the frequency domain, the convolution becomes the following multiplication:

$$ \hat{C}\left( {{{\mathbf{x}}}_{B} ,{{\mathbf{x}}}_{A}^{\left( l \right)} ,\omega } \right) = \mathop \sum \limits_{k} \widehat{{\overline{G}}}_{d} \left( {{{\mathbf{x}}}_{B} ,{{\mathbf{x}}}^{\left( k \right)} ,\omega } \right){ }\hat{\varGamma }\left( {{{\mathbf{x}}}^{\left( k \right)} ,{{\mathbf{x}}}_{A}^{\left( l \right)} ,\omega } \right) $$

(10)

Then, by solving this system of equations, the required Green’s function was obtained [13].

3 Practices in Which PSF Cannot Be Obtained from Field Data

SI by MDD, based on Eq. (10), needs two sets of data, the one is the result of SI by correlation function $C\left( {{{\mathbf{x}}}_{B} ,{{\mathbf{x}}}_{A}^{\left( l \right)} ,t} \right)$ and the other one is $\varGamma \left( {{{\mathbf{x}}}^{\left( k \right)} ,{{\mathbf{x}}}_{A}^{\left( l \right)} ,t} \right)$. The matrix equation can be solved per frequency component via such as weighted least-square inversion to obtain $\overline{G}_{d} \left( {{{\mathbf{x}}}_{B} ,{{\mathbf{x}}},t} \right)$ [8]. In practices, $C\left( {{{\mathbf{x}}}_{B} ,{{\mathbf{x}}}_{A} ,t} \right)$ can always be obtained, but the PSF is not always available. In this section we introduce two practices, in where the PSF cannot be obtained from field data.

Working face is any place where coal is extracted during a mining cycle, at the same time where is the primary disaster zone, geological hazards include caving of roof, gas outburst, pressured water outburst, etc. Figure 2a shows a top view of common longwall working face. After the coal is mined by the mining machine, it is transported to crusher by scraper conveyors. After reduced in size, the coal in crusher is loaded onto the conveyor belt and transported to the surface. Belt conveyor is installed in the belt tunnel, and it is about 0.5 m from the coal wall. The belt conveyer is in the belt tunnel from the beginning to the end, so the seismic ray can cover the whole working face [5].

Fig. 2

a Top view of coal working face. b A cross-sectional view of the stratum below the road, including subway tunnels and utility tunnel

For the foundation below the subway tunnel, accumulated deformation of soil under the subway loading should not be ignored. For example, when the deformations of foundation exceed a certain range, the safety of adjacent buildings and underground pipe networks will be endangered, causing a series of environmental and geotechnical problems [6]. Figure 2b is a cross-sectional view of subway train tunnels and utility tunnels under urban roads. Geophones are arranged in these tunnels to monitor the stratum through which seismic waves pass by, using the trains’ noise as sources. Geophones can only be arranged in these tunnels. Because it almost coincides with the distribution of passive sources, it is difficult to obtain a high signal-to-noise ratio PSF in the data of the train tunnel, so the usual MDD method is not suitable for such practice.

Figure 3a shows 10 s of the response of belt conveyor’s noise along the belt tunnel in a coal mine working face, with a total of 72 traces. The trace spacing is 10 m. Seven of them are bad traces, which are 19–21, 26 and 31–33. The belt conveyor is composed of belt and belt support, the spacing between each belt support is about 3 ~ 4 m, and the belt support is installed on the tunnel floor. It can be seen from the data that the belt conveyor is started at about 2 s. Figure 3b shows the PSF of the 36th trace of the data shown in Fig. 3a. It is found that only the autocorrelation function of the 36th trace has a relatively high signal-to-noise ratio, while the cross-correlation function with other traces is completely covered by noise, so the PSF is invalid. Therefore, the usual MDD method is not applicable to this practice.

Fig. 3

a Response along the belt tunnel of belt conveyor’s noise in a coal mine working face. b PSF of the 36th trace of the response of belt conveyor’s noise

4 SI by MDD Free from PSF

The PSF does not needed to be calculated for the proposed new MDD methods which, like multi-dimensional whitening filter, can extract PSF from the cross-correlation function $C\left( {{{\mathbf{x}}}_{B} ,{{\mathbf{x}}}_{A} ,t} \right)$ implicitly. Because the data process only uses the results of SI by cross-correlation and not need calculate PSF explicitly from seismic response, the method is named as MDD free from PSF.

Equation (7) defines the PSF in the time domain. Transforming the PSF to the frequency domain, according to

$$ \hat{\varGamma }\left( {{{\mathbf{x}}},{{\mathbf{x}}}_{A} ,\omega } \right) = \mathop \sum \limits_{i} {\hat{\text{G}}}^{in} \left( {{{\mathbf{x}}},{{\mathbf{x}}}_{S}^{\left( i \right)} ,\omega } \right){ }\left\{ {{\hat{\text{G}}}^{in} \left( {{{\mathbf{x}}}_{A} ,{{\mathbf{x}}}_{S}^{\left( i \right)} ,{\upomega }} \right)} \right\}^{*} { }\hat{W}^{\left( i \right)} \left( \omega \right) $$

(11)

Here we discuss how the alternative to the PSF is obtained from the result of SI by cross-correlation.

$\hat{C}\left( {{{\mathbf{x}}}_{B} ,{{\mathbf{x}}},\omega } \right)$ is cross-correlation of response at geophones at ${{\mathbf{x}}}_{B}$ and ${{\mathbf{x}}}$, in frequency domain. ${{\mathbf{x}}}$ is coordinate of geophones of virtual sources. Transforming the Eq. (8) to the frequency domain, according to

$$ \hat{C}\left( {{{\mathbf{x}}}_{B} ,{{\mathbf{x}}}_{A} ,\omega } \right) = \mathop \sum \limits_{i} { }\hat{G}\left( {{{\mathbf{x}}}_{B} ,{{\mathbf{x}}}_{S}^{\left( i \right)} ,\omega } \right){ }\left\{ {{\hat{\text{G}}}^{in} \left( {{{\mathbf{x}}}_{A} ,{{\mathbf{x}}}_{S}^{\left( i \right)} ,{\upomega }} \right)} \right\}^{*} { }\hat{W}^{\left( i \right)} \left( \omega \right) $$

(12)

Similarly, the $\hat{C}\left( {{{\mathbf{x}}}_{B} ,{{\mathbf{x}}},\omega } \right)$ according to

$$ \hat{C}\left( {{{\mathbf{x}}}_{B} ,{{\mathbf{x}}},\omega } \right) = \mathop \sum \limits_{i} { }\hat{G}\left( {{{\mathbf{x}}}_{B} ,{{\mathbf{x}}}_{S}^{\left( i \right)} ,\omega } \right){ }\left\{ {{\hat{\text{G}}}^{in} \left( {{{\mathbf{x}}},{{\mathbf{x}}}_{S}^{\left( i \right)} ,{\upomega }} \right)} \right\}^{*} { }\hat{W}^{\left( i \right)} \left( \omega \right) $$

(13)

$\hat{C}\left( {{{\mathbf{x}}}_{B} ,{{\mathbf{x}}},\omega } \right)$ and $\hat{C}\left( {{{\mathbf{x}}}_{B} ,{{\mathbf{x}}}_{A} ,\omega } \right)$ are cross-correlated, according to

$$ \begin{aligned} & \left\{ {\hat{C}\left( {{{\mathbf{x}}}_{B} ,{{\mathbf{x}}},\omega } \right)} \right\}^{*} \hat{C}\left( {{{\mathbf{x}}}_{B} ,{{\mathbf{x}}}_{A} ,\omega } \right) = \mathop \sum \limits_{i} {\hat{\text{G}}}^{in} \left( {{{\mathbf{x}}},{{\mathbf{x}}}_{S}^{\left( i \right)} ,\omega } \right){ }\left\{ {{\hat{\text{G}}}^{in} \left( {{{\mathbf{x}}}_{A} ,{{\mathbf{x}}}_{S}^{\left( i \right)} ,{\upomega }} \right)} \right\}^{*} \\ & \left\{ {\hat{G}\left( {{{\mathbf{x}}}_{B} ,{{\mathbf{x}}}_{S}^{\left( i \right)} ,\omega } \right)} \right\}^{*} { }\hat{G}\left( {{{\mathbf{x}}}_{B} ,{{\mathbf{x}}}_{S}^{\left( i \right)} ,\omega } \right){ }\left\{ {\hat{W}^{\left( i \right)} \left( \omega \right)} \right\}^{{ *}} \hat{W}^{\left( i \right)} \left( \omega \right) \\ & \quad = \hat{\varGamma }\left( {{{\mathbf{x}}},{{\mathbf{x}}}_{A} ,\omega } \right){ }\left\{ {\hat{G}\left( {{{\mathbf{x}}}_{B} ,{{\mathbf{x}}}_{S}^{\left( i \right)} ,\omega } \right)} \right\}^{*} { }\hat{G}\left( {{{\mathbf{x}}}_{B} ,{{\mathbf{x}}}_{S}^{\left( i \right)} ,\omega } \right){ }\left\{ {\hat{W}^{\left( i \right)} \left( \omega \right)} \right\}^{*} \\ \end{aligned} $$

(14)

The right-hand side of the above equation contains 3 components, and the first part is just the PSF. The second part is

$$ \mathop \sum \limits_{i} \left\{ {\hat{G}\left( {{{\mathbf{x}}}_{B} ,{{\mathbf{x}}}_{S}^{\left( i \right)} ,\omega } \right)} \right\}^{*} { }\hat{G}\left( {{{\mathbf{x}}}_{B} ,{{\mathbf{x}}}_{S}^{\left( i \right)} ,\omega } \right) $$

(15)

In the absence of multiples, this part has zero phase and has no effect on the total phase.

The third part is the last term on the right side of the equation, which is the conjugate of the source wavelet $\sum\nolimits_{i} {\hat{W}^{\left( i \right)} \left( \omega \right)}$. $\hat{W}^{\left( i \right)} \left( \omega \right) = \hat{\omega }^{\left( i \right)} \left( \omega \right) \left\{ { \hat{\omega }^{\left( i \right)} \left( \omega \right)} \right\}^{*}$ has zero phase, and the conjugate of them still has zero phase, which will not affect the phase of Eq. 13, according to

$$ \mathop \sum \limits_{i} \left\{ {\hat{W}^{\left( i \right)} \left( \omega \right)} \right\}^{*} = \mathop \sum \limits_{i} \hat{W}^{\left( i \right)} \left( \omega \right) $$

(16)

so it will not affect the phase of Eq. 14, but this term will have a certain impact on the total amplitude spectrum.

In general, $\left\{ {\hat{C}\left( {{{\mathbf{x}}}_{B} ,{{\mathbf{x}}},\omega } \right)} \right\}^{*} \hat{C}\left( {{{\mathbf{x}}}_{B} ,{{\mathbf{x}}}_{A} ,\omega } \right)$ is an alternative of the point spread function $\hat{\varGamma }\left( {{{\mathbf{x}}}^{\left( k \right)} ,{{\mathbf{x}}}_{A}^{\left( l \right)} ,\omega } \right)$ in terms of phase. The amplitude spectrum will be affected by the zero-phase correlation wavelet. Compared with PSF, the amplitude spectrum of $\left\{ {\hat{C}\left( {{{\mathbf{x}}}_{B} ,{{\mathbf{x}}},\omega } \right)} \right\}^{*} \hat{C}\left( {{{\mathbf{x}}}_{B} ,{{\mathbf{x}}}_{A} ,\omega } \right)$ is narrower, which is equivalent to that the PSF is filtered by the source wavelet $\sum\nolimits_{i} {\hat{W}^{\left( i \right)} \left( \omega \right)}$.

$\left\{ {\hat{C}\left( {{{\mathbf{x}}}_{B} ,{{\mathbf{x}}},\omega } \right)} \right\}^{*} \hat{C}\left( {{{\mathbf{x}}}_{B} ,{{\mathbf{x}}}_{A} ,\omega } \right)$ can be written as ${\hat{\user2{P}}}$. The inverse filter can be obtained per frequency component via weighted least-squares inversion, according to

$$ {\hat{\user2{F}}} = \left( {{\hat{\user2{P}}} + \varepsilon^{2} {{\mathbf{I}}}} \right)^{ - 1} $$

(17)

where the $\varepsilon^{2}$ is a stabilization parameter, ${{\mathbf{I}}}$ is the identity matrix. $\varepsilon^{2}$ is frequenc y dependent. Applying the inverse filter for each frequency component and transforming the result to the time domain is equivalent with the deconvolution in the time domain.

5 Numerical Example 1: Thread Source One-Side Illumination

The numerical example in this section is a general one-side illumination, the source is a thread source, such as the train [18,19,20,21] or the highway [9]. With the development of transportation infrastructure, this kind of passive source is getting more attention [22,23,24,25]. In this example, there is a certain distance between the passive source and the geophones (Line1 shown in Fig. 4). In the example in the next section, the passive sources and the geophones which as the virtual sources are almost coincident (shown in Fig. 6).

Fig. 4

Configuration of ‘virtual source method’ for far-field illumination from one-side. Two sets of geophone arrays are respectively in two tunnels (the triangles at Line1 and Line2)

Consider the configuration in Fig. 4, which consists of a set of thread sources and two sets of geophone arrays. The length of geophone array is 500 m, and the thread source is composed of a group of independent noise sources, with a total of 51 sources and trace spacing of 10 m. The source spacing is 10 m too. It is 300 m from the thread source to Line1 and 300 m from Line1 to line2.

The source time function of each noise source is represented by a random sequence of standard normal distribution, which is uncorrelated with each other. The longer the random sequence used, the higher the signal-to-noise ratio of the shot gather obtained by cross-correlation. Because it propagates in the homogeneous medium, the signal from each source point arrives at each geophone only with time delay and energy diffusion attenuation.

Figure 5a shows the result of SI by cross-correlation, and the random sequence length is 100,000. The virtual source is at the 26th geophone in Line1 (the middle trace), and this results in a great signal-to-noise ratio in the image. In the results, the source blurring is very obvious due to the one-side illumination, especially in the middle traces. Figure 5b shows the PSF of trace 26 in Line1, which is the cross-correlation function between trace 26 and all the traces in Line1. Figure 5c is the result obtained by using the usual MDD method, and compared with Fig. 5a, the source blurring completely disappears. Figure 5d is the result obtained by using MDD method free from PSF proposed in this paper; the source blurring is also well eliminated, and the signal-to-noise ratio is greater than that of Fig. 5c.

Fig. 5

a Result of SI by cross-correlation. b PSF of trace 26 in Line1. c Result of SI by usual MDD method. d Result of SI by MDD free from PSF

Fig. 6

Configuration of ‘virtual source method’ for near-field illumination from one-side. Two sets of geophone arrays are respectively in two tunnels (the triangles at Line1 and Line2)

6 Numerical Example 2: Near Field Thread Source One-Side Illumination

In Fig. 6, the thread passive source and geophones in Line1 almost coincide with each other, which simulates the belt conveyor or train source in an underground tunnel, because the geophones as virtual sources can only be installed in the tunnel. This is a common problem for all thread sources in the tunnel. Section 3 briefly describes such practices.

In this example, the thread source is composed of 51 independent point sources with intervals of 10 m. The source time function of each source is represented by a random sequence with length of 100,000, which is uncorrelated with each other. The two groups of geophones Line1 and line2 are composed of 51 geophones respectively with a trace spacing of 10 m. The distance between Line1 and line2 is 300 m. In actual, most of the coal working face width is about 300 m. This numerical example can be directly used to study the interferometry imaging of working face with belt conveyor as the source.

Figure 7a is the result of SI by cross-correlation, which if the cross-correlation function between the response of trace 26 (middle channel) of Line1 and the responses of all traces of Line2. It is obvious that there are a series of spurious events before the true event, and these spurious events are caused by one-side illumination. Compared with the source blurring in Fig. 5a, the source blurring in Fig. 7a are some spurious events. True and spurious events can be easily distinguished in the numerical synthesis example, but it is difficult to distinguish them in the actual field data. Figure 7b is the result of SI by MDD free from PSF proposed in this paper, which largely eliminates the spurious events on the seismogram.

Fig. 7

a Result of SI by cross-correlation. b Result of SI by MDD free from PSF

In the numerical simulation, the distance of geophone between the source and Line1 is an important influencing factor. If the positions of 51 source points and 51 geophones completely coincide with each other, the proportion of overlapped source points is too large for a certain geophone, and the contribution of the remaining 50 source points can be ignored. Under the above conditions, there is no source blurring in the cross-correlation results, which is inconsistent with the actual observation results. In practices, the belt conveyor or train causes vibration on the tunnel floor, and the geophone is installed on the tunnel wall, so there is a distance of 2–3 m between the source point and the geophone at the same position.

For practices where PSF cannot be obtained in the field data, the usual MDD method cannot be used to eliminate source blurring, while the MDD method free from PSF proposed in this paper can be used to solve this problem. In Fig. 7b, the spurious events still have some residuals, but the true event is clearer compared to Fig. 7a.

7 Conclusions

We propose a new MDD method free from PSF as an alternative to the usual MDD. Using this method, MDD processing can be carried out directly from the results of SI by cross-correlation. The main advantage of the new MDD method is that the source of the Green’s function obtained by the correlation method can be deblurred for practices where PSF cannot be obtained, while the conventional MDD method must calculate PSF first. In addition, for practices where PSF can be obtained from field data, the results of the new method show that the new method and the usual method show similar processing effects.

The new method is similar to a 3-D whitening filter. Each geophone in Line1 is taken as the virtual source, and virtual shot gathers is obtained by cross-correlation. The virtual shot gathers are arranged as a 3D array for whitening. MDD needs to set stabilization parameters $\varepsilon^{2}$ which can only be determined by multiple attempts. The results of SI by MDD are given by subjective evaluation, so the results obtained by different people will be slightly different.

In practices, the SI by cross-correlation and by MDD method should be used together. The research shows that both 1D deconvolution and MDD can improve the resolution of shot gather, and at the same time, the method itself may cause spurious events. There is no spurious event in the result of SI by cross-correlation, so it can be used to determine which are spurious events in the result of SI by MDD. Although the seismic interference of such sources will still face many problems in practical applications, the use of linear sources in tunnels is getting more and more attention. This new MDD method proposed in this paper provides an attractive solution.