Accelerating the multi-parameter least-squares reverse time migration using an appropriate preconditioner

Abstract

Least-squares reverse time migration has proven to be the state-of-the-art for linear imaging technique of complex subsurface structures. Assuming a variable-density acoustic medium, least-squares iterations have the potential to compensate for the artifacts caused by finite frequency of the seismic source, limited acquisition aperture, uneven illumination and parameter cross-talk. The main drawback of such an iterative imaging scheme is the substantial computational expense induced by additional modeling/adjoint steps at each iteration. To accelerate the convergence rate, we propose to leverage the variable-density pseudoinverse extended Born operator as a preconditioner. Our imaging scheme consists of two main steps. We first construct a true-amplitude extended image through Conjugate Gradient iterations with/without preconditioning. Then, by applying a 2D Radon transform, we simultaneously estimate the physical parameters from the angle-domain response using a weighted least-squares method. The second step does not involve wave propagation terms. Through numerical experiments, we show that the proposed preconditioning scheme not only largely reduces the required number of iterations to achieve a given data misfit but also significantly increases the quality of the inverted images even in presence of strong parameter cross-talk and inaccurate migration background models. This is further confirmed by analyzing the shape of the multi-parameter Hessian obtained on a model with limited size.

Appendices

Appendix

A Derivation of Hessian

The exact definition of the normal equation reads as [45]

$$ \begin{array}{@{}rcl@{}} \int \mathrm{d} \mathbf{x}_{0} \left[\begin{array}{llllllllll} \mathbb{H}_{I_{p},I_{p}}(\mathbf{x},\mathbf{x}_{0}) & \mathbb{H}_{I_{p},\rho}(\mathbf{x},\mathbf{x}_{0}) \\ \mathbb{H}_{\rho,I_{p}}(\mathbf{x},\mathbf{x}_{0}) & \mathbb{H}_{\rho,\rho}(\mathbf{x},\mathbf{x}_{0}) \end{array}\right] \left[\begin{array}{l} \zeta_{I_{p}}(\mathbf{x}_{0}) \\ \zeta_{\rho}(\mathbf{x}_{0}) \end{array}\right]= \left[\begin{array}{l} \zeta_{I_{p}}^{mig}(\mathbf{x}) \\ \zeta_{\rho}^{mig}(\mathbf{x}) \end{array}\right], \end{array} $$

(21)

where the diagonal blocks of the Hessian matrix are

$$ \begin{array}{@{}rcl@{}} \mathbb{H}_{I_{p},I_{p}}(\mathbf{x},\mathbf{x}_{0}) = \frac{\partial^{2} J (\zeta_{I_{p}},\zeta_{\rho})}{\partial \zeta_{I_{p}}(\mathbf{x})\partial \zeta_{I_{p}}(\mathbf{x}_{0})}, \mathbb{H}_{\rho,\rho}(\mathbf{x},\mathbf{x}_{0}) = \frac{\partial^{2} J (\zeta_{I_{p}},\zeta_{\rho})}{\partial \zeta_{\rho}(\mathbf{x})\partial \zeta_{\rho}(\mathbf{x}_{0})}, \end{array} $$

(22)

and the non-diagonal blocks are

$$ \begin{array}{@{}rcl@{}} \mathbb{H}_{I_{p},\rho}(\mathbf{x},\mathbf{x}_{0}) = \frac{\partial^{2} J (\zeta_{I_{p}},\zeta_{\rho})}{\partial \zeta_{I_{p}}(\mathbf{x})\partial \zeta_{\rho}(\mathbf{x}_{0})}, \mathbb{H}_{\rho,I_{p}}(\mathbf{x},\mathbf{x}_{0}) = \frac{\partial^{2} J (\zeta_{I_{p}},\zeta_{\rho})}{\partial \zeta_{\rho}(\mathbf{x})\partial \zeta_{I_{p}}(\mathbf{x}_{0})}. \end{array} $$

(23)

Following the Schwarz’s theorem, the Hessian is a symmetric matrix. The approaches described in this article cannot provide the structure of the exact Hessian in Eqs. 22 and 23 as it consists of a two-step approach (see Algorithms 1 and 2: Conjugate Gradient followed by weighted least-squares projection). To represent the Hessian at each grid point, we first generate the response to a point scatterer either in $\zeta _{I_{p}}$ or ζ_ρ (application of ${\mathscr{L}}$). Then, we migrate the resulting data to generate a subsurface offset-domain CIG (application of ${\mathscr{L}}^{T}$), and finally apply the WLS method to estimate the corresponding point spread function in terms of $\zeta _{I_{p}}$ and ζ_ρ. The preconditioner is this scheme can be applied to the subsurface offset-domain CIG (application of ${\mathscr{L}}^{\dagger }{\mathscr{L}}^{\dagger ^{T}}$) before the WLS step. Consequently, these approaches do not follow the Schwarz’s theorem, and thus cannot provide exactly a symmetric Hessian matrix, but rather approximately the Hessian.

B Alternative approach

Hou and Symes [26] expressed the pseudoinverse Born formula as a modification of RTM operator using two weighting operators in model and data spaces as

$$ \begin{array}{@{}rcl@{}} \mathcal{L}^{\dagger}_{Hou} = W_{model} \mathcal{L}^{T} W_{data}, \end{array} $$

(24)

where

$$ \begin{array}{@{}rcl@{}} W_{model} = 32 \frac{\beta_{0}}{{\rho_{0}^{3}}}\partial_{z}, W_{data} = I_{t}(I_{t}\partial_{s_{z}})(I_{t}\partial_{r_{z}}), \end{array} $$

(25)

and I_t denotes the causal indefinite time integration. Note that the weights in Eqs. 14 and 25 are exactly the same. We refer to [7] for a more detailed comparison.

To accelerate the convergence rate of the extended LSRTM in a constant-density acoustic medium, [27] reformulated the CG algorithm by replacing Euclidean norms with new weighted norms in data (${\mathscr{D}}$) and model (${\mathscr{M}}$) spaces using W_model and W_data as

$$ \begin{array}{@{}rcl@{}} \langle \delta m_{1} , \delta m_{2} \rangle_{\mathscr{M}} &=& \langle \delta m_{1} ,W_{model} \delta m_{2} \rangle, \end{array} $$

(26)

$$ \begin{array}{@{}rcl@{}} \langle \delta d_{1} , \delta d_{2} \rangle_{\mathscr{D}} &=& \langle \delta d_{1} ,W_{data} \delta d_{2} \rangle. \end{array} $$

(27)

The algorithm of weighted CG (WCG) for variable-density LSRTM is summarized in Algorithm 3. In terms of implementation, as the application of these weight operators are relatively cheap (steps 8, 9, and 12 in Algorithm 3), the computational cost of the WCG algorithm is negligibly higher than the standard CG (Algorithm 1).

C Data availibility

The datasets generated during and/or analysed during the current study are available from the corresponding author on reasonable request.