基于一阶速度-应力方程的多震源最小二乘逆时偏移

Multi-source least-squares reverse time migration based on first-order velocity-stress wave equation

最小二乘逆时偏移(Least-Square Reverse Time Migration,LSRTM)相比于常规偏移具有更高的成像分辨率、振幅保幅性及均衡性等优势,是当前研究的热点之一.然而,目前LSRTM算法大多是基于二阶常密度标量声波方程建立的,忽略了密度变化对振幅的影响,因而基于振幅匹配策略的常规LSRTM很难在变密度介质下取得保真的成像结果.一阶速度-应力方程能够很好地处理变密度介质,但简单地将一阶速度-应力方程应用到LSRTM中缺乏理论基础.为此,本文从LSRTM的正问题入手,提出了基于交错网格的一阶速度-应力方程LSRTM理论方法.首先将一阶波动方程线性化,建立了一阶方程LSRTM的目标泛函,随后推导其伴随方程,并借助伴随状态法给出了迭代更新流程,最终建立了基于一阶速度-应力方程LSRTM的理论框架.进一步,通过在相位编码LSRTM中引入随机最优化思想,极大地减小了计算量、提高了计算效率.最后,通过模型试算验证了本算法的正确性和有效性.

Compared to the conventional migration method,Least-squares reverse time migration has a lot of advantages,such as higher imaging resolution,amplitude preservation and amplitude balance and so on.Therefore,LSRTM is the focus of current research.However,current LSRTM algorithms are mostly established based on the second-order scalar constant density acoustic wave equation.They ignore the effect of density on the amplitude,so conventional LSRTM which is based on the amplitude matching strategy is difficult to obtain a fidelity imaging result in variable density medium.First-order velocity-stress wave equation is able to handle variable density of the medium.However,direct application of the first-order velocity-stress wave equation to LSRTM lacks theoretical foundation.In this paper,we propose a new LSRTM algorithm based on first order velocity-stress wave equation.First,we derive a first-order linear wave equation,and then the misfit function of first-order equation LSRTM is given based on a L2 norm.Using the adjoint-state method we obtain the adjoint equation.Finally,a theoreticalframework of our first-order velocity-stress wave equation LSRTM method is established.In addition,by introducing the stochastic optimization in phase encoding multi-source LSRTM,the amount of computation is greatly reduced and computational efficiency is improved.Numerical tests on synthetic data demonstrate the validity of the proposed method.