Subsalt imaging by target-oriented wavefield least-squares migration: A 3-D field-data example

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Appendix A

3-D conical-wave domain Hessian

In general, a 3-D surface seismic data set can be represented by a 5-D object $d({\bf x}_r,{\bf x}_s,\omega)$, with ${\bf x}_r=(x_r,y_r,z_r=0)$ and ${\bf x}_s=(x_s,y_s,z_s=0)$ being the receiver and source position, respectively, and $\omega$ being the angular frequency. Under the Born approximation (Stolt and Benson, 1986), the data can be modeled by a linear operator as follows:

$$d({\bf x}_r,{\bf x}_s,\omega) = \sum_{{\bf x}}{\omega}^2f_s(\omega) G({\bf x},{\bf x}_s,\omega) G({\bf x},{\bf x}_r,\omega) m({\bf x}),$$ (17)

where $f_s(\omega)$ is the source function; $G({\bf x},{\bf x}_s,\omega)$ and $G({\bf x},{\bf x}_r,\omega)$ are the Green's functions connecting the source and receiver position to the image point ${\bf x}=(x,y,z)$, respectively. We can transform data into the conical-wave domain by slant-stacking along the inline source axis ${x_s}$ as follows:

$$d({\bf x}_r,p_{s_x},y_s,\omega) = \sum_{x_s}W({\bf x}_r,x_s,y_s)d({\bf x}_r,x_s,y_s,\omega) {\rm e}^{i\omega p_{s_x}x_s},$$ (18)

where $W({\bf x}_r,x_s,y_s)$ is the acquisition mask operator, which contains ones where we record data, and zeros where we do not; $p_{s_x}$ is the surface ray parameter in the inline direction. The inverse transform is

$$W({\bf x}_r,x_s,y_s)d({\bf x}_r,x_s,y_s,\omega) = \vert\omega\vert \sum_{p_{s_x}} d({\bf x}_r,p_{s_x},y_s,\omega) {\rm e}^{-i\omega p_{s_x}x_s},$$ (19)

where $\vert\omega\vert$ on the right hand side of the equation is also known as the ``rho'' filter (Claerbout, 1985).

To find a reflectivity model ${\bf m}$ that best fits the observed data for a given background velocity, we can minimize a data-misfit function that measures the differences between the observed data and the synthesized data in a least-squares sense. In the point-source case, the data-misfit function is

$$F({\bf m}) = \frac{1}{2}\sum_{\omega}\sum_{{\bf x}_s}\sum_{{\bf x... ...[d({\bf x}_r,{\bf x}_s,\omega)-d_{\rm obs}({\bf x}_r,{\bf x}_s,\omega)]\vert^2,$$ (20)

where $d_{\rm obs}$ is the observed data. Substituting equation A-3 into A-4 yields

$$F({\bf m}) = \frac{1}{2}\sum_{\omega}\sum_{y_s}\sum_{{\bf x}_r}\sum_{p_{s_x}}\sum_{p_{s_x}'}\vert\omega\vert^2$$

$$\times [d({\bf x}_r,p_{s_x} ,y_s,\omega)-d_{\rm obs}({\bf x}_r,p_{s_x} ,y_s,\omega)]^{*}$$

$$\times [d({\bf x}_r,p_{s_x}',y_s,\omega)-d_{\rm obs}({\bf x}_r,p_{s_x}',y_s,\omega)] \sum_{x_s}e^{-i\omega(p_{s_x}'-p_{s_x})x_s}.$$ (21)

If the inline source axis $x_s$ is reasonably well sampled, we have $\sum_{x_s}e^{-i\omega(p_{s_x}'-p_{s_x})x_s}\approx\frac{1}{\vert\omega\vert}\delta(p_{s_x}'-p_{s_x})$, where $\delta(\cdot)$ is the Dirac delta function. Therefore, an objective function equivalent to equation A-4 in the 3-D conical-wave domain takes the following form:

$$F({\bf m}) \approx \frac{1}{2}\sum_{\omega}\vert\omega\vert\sum_{... ...bf x}_r,p_{s_x},y_s,\omega) - d_{\rm obs}({\bf x}_r,p_{s_x},y_s,\omega)\vert^2.$$ (22)

The Hessian operator in the 3-D conical-wave domain can be obtained by taking the second-order derivatives of $F({\bf m})$ (equation A-6) with respect to the model parameters:

$$H({\bf x},{\bf x}') = \sum_{\omega} \vert\omega\vert^5\sum_{y_s} ... ...rac{\partial d({\bf x}_r,p_{s_x},y_s,\omega)}{\partial m({\bf x}')}\right)^{*}.$$ (23)

When ${\bf x}={\bf x}'$, we obtain the diagonal components of the Hessian, which are also known as the subsurface illumination; otherwise, we obtain the off-diagonal components of the Hessian, which are also known as the resolution function for a given acquisition setup.

With equations A-1 and A-2, we obtain the expression of the derivative of $d$ with respect to $m$ as follows:

$$\frac{\partial d({\bf x}_r,p_{s_x},y_s,\omega)}{\partial m({\bf x})} = \sum_{x_s} {\omega}^2 W({\bf x}_r,x_s,y_s) f_s(\omega)G({\bf x},x_s,y_s,\omega)$$

$$\times G({\bf x},{\bf x}_r,\omega) {\rm e}^{i\omega p_{s_x}x_s}.$$ (24)

Substituting equation A-8 into equation A-7 yields the expression for each component of the Hessian matrix in the 3-D conical-wave domain:

$$H({\bf x},{\bf x}') = \sum_{\omega} \vert\omega\vert^5\sum_{y_s} \sum_{p_{s_x}} \sum_{{\bf x}_r} G({\bf x},{\bf x}_r,\omega) G^{*}({\bf x}',{\bf x}_r,\omega)$$

$$\sum_{x_s} W({\bf x}_r,x_s ,y_s) f_s (\omega) G ({\bf x} ,x_s ,y_s,\omega) {\rm e}^{ i\omega p_{s_x} x_s }$$

$$\sum_{x_s'} W({\bf x}_r,x_s',y_s) f_s^{*}(\omega) G^{*}({\bf x}',x_s',y_s,\omega) {\rm e}^{-i\omega p_{s_x} x_s'}.$$ (25)

Subsalt imaging by target-oriented wavefield least-squares migration: A 3-D field-data example