Time Domain Waveform Inversion for theQModel Based on the First-Order Viscoacoustic Wave Equations

Abstract

Propagating seismic waves are dispersed and attenuated in the subsurface due to the conversion of elastic energy into heat. The absorptive property of a medium can be described by the quality factorQ. In this study, the first-order pressure-velocity viscoacoustic wave equations based on the standard linear solid model are used to incorporate the effect ofQ. For theQmodel inversion, an iterative procedure is then proposed by minimizing an objective function that measures the misfit energy between the observed data and the modeled data. The adjoint method is applied to derive the gradients of the objective function with respect to the model parameters, that is, bulk modulus, density, andQ-related parameterτ. Numerical tests on the crosswell recording geometry indicate the feasibility of the proposed approach for theQanomaly estimation.

Viscoelastic and viscoacoustic modelling using the Lie product formula

“…The 2D viscoacoustic and viscoelastic wave equations are expressed by several researchers using the standard linear solid (SLS), also called the Zener model (Zhang and Gao, 2016;Hong-Yong and Yang, 2012;Zhu et al, 2013). According to Hao and Greenhalgh (2019), the SLS model is a classic mechanical representation of anelastic behaviour and it comprises a series combination of a spring and a Kelvin-Voigt solid or a parallel combination of a spring and a Maxwell solid.…”

“…and the vector u = (v x , v z , σ, r) T , where σ is the pressure field and r is the memory variable (Zhang and Gao, 2016). The viscoelastic and viscoacoustic wave equations in the TTI medium can be calculated by rotating the system of the VTI medium locally.…”

“…The 2D viscoacoustic and viscoelastic wave equations are expressed by several researchers using the standard linear solid (SLS), also called the Zener model (Zhang and Gao, 2016; Hong‐Yong and Yang, 2012; Zhu et al. , 2013).…”

“…Methods in the second category use relaxation functions to describe the viscoelastic constitutive relationship. The standard linear solid (SLS) and generalized standard linear solid are widely used as viscoelastic models to introduce memory variables into propagation modelling (Blanch et al, 1995;Carcione, 1995;Zhang and Gao, 2016;Hong-Yong and Yang, 2012;Robertsson et al, 1994;Zhu et al, 2013).…”