1.6 Multi-focusing diffraction imaging (MFDI)

MFDI is a novel temporal displacement correction method that effectively describes diffraction events, with the ideal sum of diffracted events and the


Table 1.

The relationship between seismic wave frequency and other parameters.

## Advance Wave Modeling and Diffractions for High-Resolution Subsurface Seismic Imaging DOI: http://dx.doi.org/10.5772/intechopen.81164

attenuation of specular reflections allowing the creation of an image mainly comprising diffraction energy. The time correction, which is based on the multiple focus method, depends on two parameters, the emergence angle and the radius of curvature of the diffracted wavefront. The above parameters are calculated from the seismic traces of the pre-stack [19]. The result of the IFM is therefore a highresolution full-azimuth seismic image that comprises optimally stacked diffraction events. The diffraction section contains important data on local heterogeneities and discontinuities in the geology of the subsoil, which can be used to improve horizontal drilling and determine the optimal location of exploration wells.

The multi-focus diffraction imaging makes it possible to:


#### 1.7 Finite difference modeling

Finite difference methods (FDM) are extensively used in seismic modeling and migration. In this chapter, a conventional FDM model was used for modeling, with model input being velocity and density values, and the model produces seismic data. FDM are numerical methods for solving differential equations by approximating them with differential equations, in which finite differences approach derivatives. In seismic wave modeling, FD methods are used to propagate the wave in the subsoil. This method has no immersion limitation and produces all events associated with the wave equation such as multiple reflection, head waves and elastic wave equation, anisotropic effects, and wave conversion data [21]. Therefore, the modeling of the F-D wave equations is the ideal way to produce the synthetic seismic data. However, the ultimate goal of migration is to get the image of the real earth using seismic data, which is difficult to test the accuracy of the migration methods with the desired results. In case of seismic inversion, the entry includes the traces and the output of the structural image. For this purpose, the quantities below must be calculated [22]:


For each source or receiver, the above quantities must satisfy the equations [22].

$$
\left(\frac{\partial \mathbf{r}}{\partial \mathbf{x}}\right)^2 + \left(\frac{d\mathbf{r}}{d\mathbf{x}}\right)^2 = \frac{1}{v^2(\mathbf{x}, z)}\tag{1}
$$

$$
\sin \mathcal{Q} = v \frac{d\tau}{d\mathbf{x}} \tag{2}
$$

$$\frac{\partial \sigma}{\partial \mathbf{x}} \frac{\partial \mathbf{r}}{\partial \mathbf{x}} + \frac{\partial \sigma}{\partial \mathbf{z}} \frac{\partial \mathbf{r}}{\partial \mathbf{z}} = \mathbf{1} \tag{3}$$

$$\frac{\partial \beta}{\partial \mathbf{x}} \frac{\partial \mathbf{r}}{\partial \mathbf{x}} + \frac{\partial \beta}{\partial \mathbf{z}} \frac{\partial \mathbf{r}}{\partial \mathbf{z}} = \mathbf{1} \tag{4}$$

$$\frac{\partial}{\partial \mathbf{z}} \left( \frac{\partial \boldsymbol{\beta}}{\partial \mathbf{x}} \right) + \frac{\partial}{\partial \mathbf{x}} \left[ \mu(\mathbf{x}, z) \frac{\partial \boldsymbol{\beta}}{\partial \mathbf{x}} \right] = \mathbf{0} \tag{5}$$

˙ ˆ�<sup>1</sup> where <sup>∂</sup><sup>τ</sup> v(x, z) is the velocity and <sup>μ</sup>ðx; <sup>z</sup>Þ ¼ <sup>∂</sup><sup>τ</sup> . <sup>∂</sup><sup>x</sup> <sup>∂</sup><sup>z</sup>

Eq. (1) is the Eikonal equation, Eqs. (3) and (4) are derived from Pusey and Vidale [23] and Eq. (5) follows from Eq. (3).

### 1.8 Plane-wave destructors (PWD)

Plane wave destruction filter is derived from the local plane wave model for characterizing the seismic data. This filter operates in the time domain (T-X), such as the time distance, and can be extended to the frequency domain. PWD is constructed using an imbedded finite difference scheme for the local plane wave equation as described [24].

For the characterization of several plane waves, it is possible to flow several filters similar to that of the following equation:

$$A(Z\_x) = \left(\mathbf{1} - \frac{Z\_x}{Z\_1}\right)\left(\mathbf{1} - \frac{Z\_x}{Z\_2}\right)\dots\left(\mathbf{1} - \frac{Z\_x}{Z\_N}\right) \tag{6}$$

where Z1, Z2, …, ZN are the zeros of the polynomial. The Taylor series method (assimilating the coefficients of the expansion of the Taylor series to the zero frequency) gives the expression

$$B\_3(Z\_t) = \frac{(\mathbf{1} - \sigma)(\mathbf{2} - \sigma)}{\mathbf{1}\mathbf{2}} Z\_t^{-1} + \frac{(\mathbf{2} + \sigma)(\mathbf{2} - \sigma)}{\mathbf{6}} + \frac{(\mathbf{1} + \sigma)(\mathbf{2} + \sigma)}{\mathbf{1}\mathbf{2}} Z\_t \tag{7}$$

For a three-point centered filter B3 (Zt)

$$\begin{split} B\_5(Z\_t) &= \frac{(1-\sigma)(2-\sigma)(3-\sigma)(4-\sigma)}{1680} Z\_t^{-2} + \frac{(4-\sigma)(2-\sigma)(3-\sigma)(4+\sigma)}{420} Z\_t^{-1} \\ &+ \frac{(4-\sigma)(3-\sigma)(3+\sigma)(4+\sigma)}{280} + \frac{(4-\sigma)(2+\sigma)(3+\sigma)(4+\sigma)}{420} Z\_t \\ &+ \frac{(1+\sigma)(2+\sigma)(3+\sigma)(4+\sigma)}{1680} Z\_t^2 \end{split} \tag{8}$$

For a five-point centered filter B5 (Zt), the derivation of Eqs. (7) and (8) can be found in Fomel [24].

The filter used in the present work is a modified version of filter A (Zt, Zx):

$$\mathbf{C}(\mathbf{Z}\_l, \mathbf{Z}\_\mathbf{x}) = \mathbf{A}\left(\mathbf{Z}\_l, \mathbf{Z}\_\mathbf{x}\right) \mathbf{B}\left(\frac{\mathbf{1}}{\mathbf{Z}\_l}\right) = \mathbf{B}\left(\frac{\mathbf{1}}{\mathbf{Z}\_l}\right) - \mathbf{Z}\_\mathbf{x} \mathbf{B}(\mathbf{Z}\_l) \tag{9}$$

This filter avoids the requirement of a polynomial partition. In the case of the three-point filter Eq. (7) and the two-dimensional filter Eq. (7), there are six coefficients consisting of two columns; in each column there are three coefficients and the second column is an inverse copy of the first one. However, the decomposition algorithm is significantly more expensive than the FD. This algorithm is extended to Song's anisotropic wave propagation in 2013 by involving the Eigen function rather than rows and columns of the original extrapolation matrix [25] (Figure 2).

Advance Wave Modeling and Diffractions for High-Resolution Subsurface Seismic Imaging DOI: http://dx.doi.org/10.5772/intechopen.81164

Figure 2.

(a) Snapshot of a wave field in a smooth velocity model calculated using the fourth order finite difference method; and (b) the lower rank approximation of the wave field in the same smooth velocity model [26].

#### 1.9 Slope estimation

Slope estimation is a necessary step in applying the FD plane-wave filters to real data [27], although estimating dissimilar dual slopes, σ<sup>1</sup> and σ2, in the available data is complicated than estimating a single slope [24].

The regularization condition should thus be applied to both Δσ<sup>1</sup> and Δσ<sup>2</sup> as follows:

$$
\epsilon D \Delta \sigma\_1 \approx 0 \tag{10}
$$

$$
eD\Delta\sigma\_2 \approx 0\tag{11}
$$

The solutions of the above equations depend on the primary values of slop. 1 and slop. 2, which should not be equal, but can be prolonged to the numerical equation with respect to the grid number of the dataset. However, this equation is used here to calculate slopes for the given data set. In the current study, we used a reformed and better version of the plane wave destruction method for seismic diffraction parting based on Claerbout [28].

#### 1.10 Diffraction separation methods from seismic full-wave data

One of the best practices and methods of diffraction preservation is the plane wave destruction (PWD) filter initially introduced by Claerbout [28] for the description of seismic images using local plane wave superposition. This PWD filter is based on the plane-wave differential equation, after the original wavedestroying filter plane with the same approximation showed poor performance when applied to spatially aliased data in comparison with other filters. Frequencydistance prediction error [16]. Planar Wave Destruction Filter, which can be considered as a time-distance (TX) analog of the frequency-distance (FX) prediction error filter, derived from a local plane wave model is used to characterize the seismic data [24]. Unfortunately, the first experiments in applying flat-wave destruction for spatially-aliased data interpolation [28] have shown poor results compared to standard frequency distance prediction error (FX) filters [16]. A workflow that uses plane wave destruction for diffraction imaging is shown below. Flat-wave destruction of common shift data may have difficulty in extracting diffractions in regions with complex geological and velocity variations [11] (Figure 3).
