**1. Introduction**

Conventional migration methods aim to create structural images of subsurface. Advances in the true-amplitude migration method further generate subsurface images with an amplitude approximate to the reflectivity of the subsurface reflectors. For acoustic cases with varying velocity and density, reflectivity is caused by the velocity and impedance changes across the interfaces. Acoustic impedance of the subsurface can be used for the direct interpretation of volume information, such as lithology and pore fill, allowing for target delineation. Those inferred rock properties can provide additional information for geologic interpretation and reservoir characterization, which may not be available from conventional seismic images [1,2]. Furthermore, relating acoustic impedance derived from seismic data to formation properties could have a significant impact on defining new potential drilling locations and optimizing well placement [3].

Earlier studies on ray+Born migration/inversion [4–7] solved the forward problem based on the Born approximation using Green's functions computed by ray theory, and implemented linearized inversion to recover the perturbed model parameters (velocity or acoustic impedance perturbation in acoustic cases; P-wave and S-wave impedance perturbations and density in elastic cases) from the observed data. However, ray-tracing based asymptotic theory is fundamentally flawed in simulating low frequency wave propagation, which is critical for an accurate estimation of media properties with blocky structures [8]. Bleistein et al. [9] extended the method by using more general Green's functions, other than the asymptotic forms. Zhang et al. [10] further developed the amplitude-preserving RTM to predict both impedance and velocity perturbations from angle-domain common-image gathers. However, RTM angle gathers for the purpose of impedance inversion can be computationally expensive. Here, we propose a modified inverse scattering imaging condition

**Citation:** Liang, H.; Zhang, H.; Liu, H. Estimation of Relative Acoustic Impedance Perturbation from Reverse Time Migration Using a Modified Inverse Scattering Imaging Condition. *Appl. Sci.* **2023**, *13*, 5291. https://doi.org/10.3390 app13095291

Academic Editor: Jianbo Gao

Received: 8 March 2023 Revised: 14 April 2023 Accepted: 20 April 2023 Published: 23 April 2023

**Copyright:** © 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

for RTM, in order to output the relative impedance perturbation from the stacked images without explicitly computing angle gathers.

In this paper, we first give an overview of model parameters estimation (relative impedance and velocity perturbations) from the observed data using common-shot RTM, in accordance with Zhang et al. [10]. Then we derive the modified inverse scattering imaging condition for the relative impedance perturbation estimation for the acoustic case with variable velocity and density. The conventional inverse scattering imaging condition was designed to reduce RTM artifacts caused by the correlation of source and receiver wavefields propagating in the same direction [11], such as backscattered and turning wave energy. The proposed modified imaging condition employs an exponential weighting function to the conventional inverse scattering imaging condition to select near-angle reflections, from which the relative impedance perturbation can then be estimated. Finally, we validate the proposed method on synthetic examples.
