Seismic wave illumination analysis

Seismic-wave illumination becomes increasingly problematic in regions with rugged topography and complex geological structure with severe lateral velocity variations. In these hostile cases, we think that seismic-wave illumination is much more important than seismic-data regularization. Without enough illumination for the target reflectors, regular seismic data can not guarantee a quality image.

Seismic-wave illumination is related to the macro-velocity model and the acquisition configuration, both of which are embodied in the Green's function. In fact, seismic-wave illumination analysis inherently is an issue of seismic-wave propagation and observation in the presence of complex velocity structure. Whether a seismic wave reaches a target reflector and whether the reflected wave is received are both important.

() analyze seismic-wave illumination with Beamlet Propagators. With directional illumination maps, the illumination of a reflector is demonstrated. (, ) discuss how the imaging resolution and amplitude are affected by the acquisition geometries with focal beams: emission-focusing and detection-focusing. However, neither methods deals with the compensation for illumination deficiency from the perspective of inverse imaging.

In least-squares inversion theory, the Hessian matrix--the second-order derivatives of the wavefield about the perturbation of a physical parameter--is given. The Hessian matrix is closely related to the seismic-wave illumination of a target reflector.

The two important issues of seismic-wave illumination analysis are (1) compensating for illumination deficiencies and (2) evaluating acquisition patterns and guiding their design.

Seismic-wave migration imaging can be represented by the following matrix equation:  

$$\textbf{m}=\left( L^{*}\right) ^{T}\textbf{d},$$ (28)

where  

$$\textbf{d}=\left(d_{x_{1}},d_{x_{2}},...,d_{x_{n}} \right)^{T},$$ (29)

and  

$$\textbf{m}=\left(m_{x_{1}},m_{x_{2}},...,m_{x_{l}} \right)^{T},$$ (30)

 

$$\left( \textbf{L}^{*}\right) ^{T}=\left( \begin{array} {cccc... ...\ L^{*}_{l1}&L^{*}_{l2}&\cdots &L^{*}_{ln}\end{array} \right) .$$ (31)

The index l is the number of imaging points or scattering points along the in-line direction in the Z^th layer; n is the number of shot-receiver pairs; L^*_ij are the complex amplitudes of the conjugate Green's functions corresponding to the imaging points. In the matrix $\left(\textbf{L}^{*} \right)^{T}$, each row is a Green's function for an imaging point in the Z^th layer. In fact, equation () is the Kirchhoff integral migration formula expressed in matrix form.

However, if seismic-wave illumination is considered, the concept of double focusing (emission focusing and detection focusing) should be introduced into the general migration-imaging formula (), following Berkhout's notation (, ):  

$$\textbf{D}\textbf{L}^{U}R\textbf{L}^{D}\textbf{S}=\textbf{d}^{obs}.$$ (32)

The matrix formula stands for the emission of a wavefield from the source S and the detection by the receivers D; meanwhile the energy of the wavefield propagates downward to the reflector R with an ideal propagator $\textbf{L}^{D}$, and is reflected back to the surface; $\textbf{L}^{U}$ is an ideal upward propagator.

Defining $\textbf{F}^{U}=\left( \textbf{D}^{*}\right)^{T} \left( \textbf{L}^{*U}\right)^{T}$ and $\textbf{F}^{D}=\left( \textbf{L}^{*D}\right)^{T}\left( \textbf{S}^{*}\right)^{T}$ gives us the formulae for detection focusing and emission focusing, respectively. Together, they represent the illumination of a point on a reflector.

We will analyze the seismic-wave illumination of a target reflector with the local Hessian matrix and compare this with the double-focusing approaches.