Acoustic–Elastic Coupled Equations for Joint Elastic Imaging of TS and Sparse OBN/OBS Data

Abstract

Elastic wave imaging with ocean-bottom four-component (4C) seismic data offers unique advantages in offshore oil and gas exploration, but for sparse ocean-bottom node/ocean-bottom seismometer (OBN/OBS) 4C seismic data, the problems of imaging acquisition footprints, poor phase continuity, and low signal-to-noise (S/N) persist. Pressure data acquired by towed streamer (TS) are dense and can be used in elastic imaging of sparse OBN/OBS data using acoustic–elastic coupled equations (AECEs). Based on the AECEs, we propose a novel method for elastic wave joint imaging of dense TS and sparse OBN/OBS seismic data, including joint elastic reverse-time migration (J-ERTM) and joint elastic least-squares reverse-time migration (J-ELSRTM), to solve the above problems in elastic imaging with sparse OBN/OBS data. J-ERTM of TS and OBN/OBS data can address the problem of imaging acquisition footprints, but with low S/N ratio. J-ELSRTM uses Hessian information and can theoretically suppress the negative influences caused by sparse acquisition, thereby obtaining better images than J-ERTM with higher S/N ratio. The results of synthetic and field data experiments show that our method is of prime importance for addressing the elastic wave imaging problems of sparse OBN/OBS data in deep-water oil and gas seismic exploration.

Appendices

Appendix A: Improved AECEs in 2D Isotropic Media

Yu et al. (2016) give the first-order AECEs in 2D isotropic media as follows:

$$ \left\{ \begin{gathered} \rho \frac{{\partial^{2} u_{x} }}{{\partial t^{2} }} = \frac{{\partial \tau_{xz}^{s} }}{\partial z} + \frac{{\partial \left( {\tau_{xx}^{s} - p_{s} } \right)}}{\partial x} \hfill \\ \rho \frac{{\partial^{2} u_{z} }}{{\partial t^{2} }} = \frac{{\partial \tau_{xz}^{s} }}{\partial x} + \frac{{\partial \left( {\tau_{zz}^{s} - p_{s} } \right)}}{\partial z} \hfill \\ p_{s} = - \left( {\lambda + \mu } \right)\left( {\frac{{\partial u_{x} }}{\partial x} + \frac{{\partial u_{z} }}{\partial z}} \right) \hfill \\ \tau_{xx}^{s} = \mu \left( {\frac{{\partial u_{x} }}{\partial x} - \frac{{\partial u_{z} }}{\partial z}} \right) \hfill \\ \tau_{zz}^{s} = - \mu \left( {\frac{{\partial u_{x} }}{\partial x} - \frac{{\partial u_{z} }}{\partial z}} \right) \hfill \\ \tau_{xz}^{s} = \mu \left( {\frac{{\partial u_{x} }}{\partial z} + \frac{{\partial u_{z} }}{\partial x}} \right) \hfill \\ \end{gathered} \right.{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} . $$

(9)

In this expression, ${\mathbf{u}} = \left( {u_{x} ,u_{z} } \right)^{T}$ is the particle velocity, $\rho$ is the density, and ${{\varvec{\uptau}}} = \left( {\tau_{xx} ,\tau_{zz} ,\tau_{xz} } \right)^{T}$ is the stress, $p_{s} {\kern 1pt} {\kern 1pt}$ represent the synthesized pressure. However, there is some redundancy in the original 2D AECEs. Thus, we define

$$ \tau_{n}^{s} = \tau_{xx}^{s} = - \tau_{zz}^{s} ,{\kern 1pt} {\kern 1pt} \tau_{s}^{s} = \tau_{xz}^{s} {\kern 1pt} {\kern 1pt} . $$

(10)

Here, $\tau_{n}^{s}$ and $\tau_{s}^{s}$ are the redefined normal and deviatoric stress components in this paper.

The improved 2D AECEs can be given by

$$ \left\{ \begin{gathered} \rho \frac{{\partial^{{2}} u_{x} }}{{\partial t^{2} }} = \frac{{\partial \tau_{s}^{s} }}{\partial z} + \frac{{\partial \left( {\tau_{n}^{s} - p_{s} } \right)}}{\partial x} \hfill \\ \rho \frac{{\partial^{2} u_{z} }}{{\partial t^{2} }} = \frac{{\partial \tau_{s}^{s} }}{\partial x} - \frac{{\partial \left( {\tau_{n}^{s} + p_{s} } \right)}}{\partial z} \hfill \\ p_{s} = - \left( {\lambda + \mu } \right)\left( {\frac{{\partial u_{x} }}{\partial x} + \frac{{\partial u_{z} }}{\partial z}} \right){\kern 1pt} {\kern 1pt} \hfill \\ \tau_{n}^{s} = \mu \left( {\frac{{\partial u_{x} }}{\partial x} - \frac{{\partial u_{z} }}{\partial z}} \right) \hfill \\ \tau_{s}^{s} = \mu \left( {\frac{{\partial v_{x} }}{\partial z} + \frac{{\partial v_{z} }}{\partial x}} \right) \hfill \\ \end{gathered} \right.. $$

(11)

The improved AECEs eliminate redundant terms, which improves the calculation efficiency. The new equations can be used for elastic wave imaging of ocean-bottom multicomponent data.

Appendix B: J-ELSRTM of TS and OBN/OBS Data

Based on the AECEs, the misfit function, the adjoint equations, and the adjoint sources are defined in Eqs. (6)–(8), respectively. RTM imaging can then be derived:

$$ {\mathbf{m}}_{{{\text{mig}}}} = - \left( {\frac{{\partial {\mathbf{S}}}}{{\partial {\mathbf{m}}}}{\mathbf{w}}} \right)^{T} {{\varvec{\Psi}}}{\kern 1pt} {\kern 1pt} {\kern 1pt} . $$

(12)

According to optimization theory, the RTM image m_mig and the reflectivity model m_ref satisfy the least-squares Newton equation when the minimum of the objective function (6) is obtained:

$$ \left( {{\mathbf{L}}^{T} {\mathbf{L}}} \right){\mathbf{m}}_{{{\text{ref}}}} = {\mathbf{m}}_{{{\text{mig}}}} {\kern 1pt} {\kern 1pt} , $$

(13)

where L represents the first-order Born operator, which is the first partial derivative of seismic data with respect to the parameters, while ${\mathbf{L}}^{T} {\mathbf{L}}$ represents the Hessian operator, which is the second-order partial derivative of E(m) with respect to the parameters.

The Hessian operator ${\mathbf{L}}^{T} {\mathbf{L}}$ in the J-ELSRTM of TS and OBN/OBS seismic data simultaneously contains the pressure and displacement terms:

$$ {\mathbf{L}}^{T} {\mathbf{L}}{ = }{\mathbf{L}}_{u}^{T} {\mathbf{L}}_{u} + {\mathbf{L}}_{p}^{T} {\mathbf{L}}_{p} {\kern 1pt} {\kern 1pt} . $$

(14)

${\mathbf{L}}_{u} ,{\kern 1pt} {\kern 1pt} {\mathbf{L}}_{p}$ are the sensitivity kernels with respect to displacement and pressure. This illustrates that J-ELSRTM of TS and OBN/OBS seismic data is theoretically more likely to take into account the influences of the pressure component on updating parameters. Since the Hessian matrix is severely ill conditioned, direct calculation of the reverse Hessian operator is not feasible. As a result, we plan to take advantage of the truncated Gauss–Newton algorithm to include Hessian information indirectly into multicomponent imaging, using the following algorithm:

Algorithm

Set ${\mathbf{x}}_{0} = 0$ and ${\mathbf{r}}_{0} = {\mathbf{H}}\left( {\mathbf{m}} \right){\mathbf{x}}_{0} - {\mathbf{b}},{\kern 1pt} {\kern 1pt} {\kern 1pt} {\mathbf{p}}_{0} = - {\mathbf{r}}_{0} ,{\kern 1pt} {\kern 1pt} k = 0$ While ${\mathbf{r}}_{k} \ne 0$ do $\begin{gathered} \alpha_{k} = \frac{{{\mathbf{r}}_{k}^{T} {\mathbf{r}}_{k} }}{{{\mathbf{p}}_{k}^{T} {\mathbf{H}}\left( {\mathbf{m}} \right){\mathbf{p}}_{k} }} = \frac{{{\mathbf{r}}_{k}^{T} {\mathbf{r}}_{k} }}{{{\mathbf{p}}_{k}^{T} \left[ {{\mathbf{L}}^{T} \left( {{\mathbf{Lp}}_{k} } \right)} \right]}}{\kern 1pt} {\kern 1pt} , \hfill \\ x_{k + 1} = x_{k} + \alpha_{k} {\mathbf{p}}_{k} , \hfill \\ {\mathbf{r}}_{k + 1} = {\mathbf{r}}_{k} + \alpha_{k} {\mathbf{H}}\left( {\mathbf{m}} \right){\mathbf{p}}_{k} = {\mathbf{r}}_{k} + \alpha_{k} \left[ {{\mathbf{L}}^{T} \left( {{\mathbf{Lp}}_{k} } \right)} \right], \hfill \\ \beta_{k + 1} = \frac{{{\mathbf{r}}_{k + 1}^{T} {\mathbf{r}}_{k + 1} }}{{{\mathbf{r}}_{k}^{T} {\mathbf{r}}_{k} }}, \hfill \\ {\mathbf{p}}_{k + 1} = - {\mathbf{r}}_{k + 1} + \beta_{k + 1} {\mathbf{p}}_{k} , \hfill \\ k = k + 1, \hfill \\ \end{gathered}$ End while Return

In this algorithm, ${\mathbf{x}}_{0}$ denotes the initial LSM images, which is set to be zero. ${\mathbf{H}}$ is the Hessian operator, and ${\mathbf{b}}$ is the RTM result. ${\mathbf{r}}_{0}$ and ${\mathbf{p}}_{0}$ are intermediary variables; $\alpha$ and $\beta$ are the step length. In each iteration, the key calculations are spent in performing the demigration ${\mathbf{Lp}}_{k}$ and migration ${\mathbf{L}}^{T} \left( {{\mathbf{Lp}}_{k} } \right)$ processes. Either demigration or migration processes can be realized with only one-time wavefield simulation. When the minimum of the 4C misfit function is achieved, J-ELSRTM images can be obtained.