Study On Frequency-dependent Characteristics And AVO Inversion Based On Spherical-wave Reflection Coefficient

Based on infinite half space medium with one flat interface,this paper mainly investigated the frequency-dependent characteristics of spherical wave reflection coefficient in acoustic,elastic and viscoelastic medium,and demonstrated the feasibility of parameter inversion using the frequency-dependent phenomenon.Firstly,different integration algorithms of calculating spherical wave reflection coefficient were reviewed and compared.In the case of flat interface,the single-frequency spherical wave reflection coefficients calculated by the regularized weighted integration method and the effective reflection coefficient method are identical,but the effective reflection coefficient method can adapt to the situation of curved interface and anisotropic medium.Compared with the amplitude and phase estimation method and the frequency domain superposition method,the semi-analytical method is more efficient to obtain the band-limited reflection coefficient.Due to the interference of refracted waves,the single-frequency spherical wave reflection coefficient is oscillating after the critical angle,while the band-limited spherical wave reflection coefficient is smooth.The single-frequency spherical wave reflection coefficient is the weighted superposition of all angular plane waves,while the band-limited reflection coefficient is only the weighted superposition of plane waves in a small range near the incident angle.Conventional AVO inversion usually requires reflection data of multiple offsets,while frequency-dependent inversion based on spherical wave reflection coefficient only need one offset data to estimate medium parameters.Through sensitivity and resolution matrix analysis,it was found that small angle information can invert density better,and large angle information can invert velocity better.In the case of v₁<v₂,there is a strong coupling relationship between velocity and density,therefore,it is difficult to simultaneously estimate velocity and density using only small angle data.It was suggested to first invert velocity using large angle data,and then to invert density using small angle data.Generally,the density information is already included in small angle data;the main function of the large angle data is to decouple the velocity and density.Using the reflected record including critical angle data can obviously improve the inversion result because the existence of the refracted wave can enrich the frequency-dependent information.In the case of v₁>v₂,density parameters can be exactly estimated using only small offset data,and the record including large offset data has little influence on the inversion results.The frequency-dependent inversion combining multiple offsets can make the inversion result more concentrated and the error will become smaller.Starting from the general solution of elastic wave equation,the integral expression of potential function is derived by using the boundary condition in elastic medium.The displacement components calculated by numerical integration method and finite difference method are compared for the first time.Then,the problem of infinite reflection coefficient was well solved by adding the P-wave and S-wave together.The low-frequency limit value of spherical wave reflection coefficient was derived,when KR?K is wave number and R is propagation distance?is close to zero.Numerical analysis also showed that the low-frequency limit value was independent of the P-wave velocity in the lower medium.Then,an attempt using frequency-dependent information of elastic P-wave was carried out to invert parameters.Frequency-dependent inversion of reflected P-wave with a single offset can achieve the accurate estimation of three parameters.With the addition of multi-offset data,the accuracy of frequency-dependent inversion is significantly improved.Finally,the spherical wave reflection response characteristics of four AVO models were investigated in viscoelastic media.For all four types of AVO models,the waveforms of plane reflection wave are always symmetric,while the waveforms of spherical reflection wave are obviously asymmetric,due to the frequency-dependent phenomenon.For class I and II AVO models,the amplitude and phase of plane wave reflection coefficient are obviously abrupt around the critical angle,while the spherical wave reflection coefficients are smooth near the critical angle.Obvious differences can be observed at both very low frequencies and near critical angles.For class III and IV AVO models with no critical angle,the plane wave reflection coefficient and the spherical wave reflection coefficient are both slowly varying,and obvious differences can be observed only at very low frequencies.