From data to model space

We follow the procedure outlined by Bernasconi et al. (1994). P-wave isotropic point sources and receivers are located on the surface of a medium characterized by small 2D deviations of the elastic parameters from a known uniform background. We compute synthetic seismograms using the linearized scatter patterns of a point diffractor (Wu and Aki, 1985) and the Born approximation: small reflection coefficients make multiple reflections negligible. The reflectivity is expressed as a linear combination of elastic perturbations. The chosen parameter set is P-impedance relative perturbations $\Delta Ip / Ip$, S-impedance relative perturbations $\Delta Is / Is$ and density relative perturbations $\Delta \rho / \rho$.The input data is a set of common shot gathers. They form a cube (data space) whose axes are

• t time,
• x_s source coordinate,
• x_h offset coordinate.

A Fourier transformation over source and offset coordinate leads to a cube in the domain

• t time,
• k_s source wavenumber,
• k_h offset wavenumber.

Transformation to the $(\omega, k_{x}, k_{z})$ domain (model domain) is accomplished through diffraction tomography techniques (Wu and Toksöš, 1987). To illustrate the basic principle of diffraction tomography we consider an infinite medium and an incident monochromatic plane wave with wavenumber ${\bf k}_{i}$. The reflected plane wave ${\bf k}_{r}$ depends upon only one Fourier component of the medium (Figure 1).

Let ${\bf k}_{m}$ be the vector of medium wavenumbers (k_x, k_z). The relation between the medium wavenumber and the incident and reflected waves is

 

$${\bf k}_{m} = {\bf k}_{r} - {\bf k}_{i}.$$ (1)

A reflected plane wave due to an incident plane wave illuminates only one point in the model space (k_x, k_z): this is a Bragg resonance effect. Therefore the mapping of data space (t, k_s, k_h) into the model space $(\omega, k_{x}, k_{z})$ is

 

k_x = k_s, (2)

 

$$k_{z} = \sqrt{ \left({\omega \over v_{r}}\right) ^{2} - k_{h... ...t{ \left({\omega \over v_{i}}\right) ^{2} - (k_{h}-k_{s})^{2}},$$ (3)

where v_r and v_i are the velocities of the reflected and incident waves, respectively.

 

diff-tomog Figure 1 Diffraction Tomography: the Bragg effect. A reflected and an incident monochromatic plane wave enlighten only one sinusoidal component of the medium.

Given k_x and k_z, we solve equations (2) and (3) to obtain a value of $\omega$ for each value of k_h. The data are migrated through a t-$\omega$ transform. The t-$\omega$ transform is evaluated with a DFT. This operation is performed one point at a time in the (k_x,k_z) domain, so we collect the contributions to the reflected field of a single sinusoidal perturbations at a time.

Spectral coverage is affected by cable length (angular aperture) and finite bandwidth. With surface acquisition, the horizontal wavenumbers are not recoverable (Figures 2 and 3) as they fall in the non-observable part of the spectrum (null space).

 

kxkz-domain Figure 2 Coverage of the model spectrum with surface acquisition and limited bandwidth from $\omega_{min}$ to $\omega_{max}$ (Wu and Toksöz, 1987). The angular aperture is linked to the the maximun observation angle. The background velocity is v.

 

mod-spec Figure 3 Effects of the limited angular aperture and bandwidth on the spectrum and on the model. Horizontal variations in the model are unrecoverable with surface acquisition as they fall in the non-observable part of the spectrum.