横向各向同性介质和斜方介质各向异性参数的约束条件

Thomsen提出的横向各向同性(TI)介质各向异性参数(α0、β0、ε、δ、γ)是各向异性理论研究和实际资料处理中的常用参数,Thomsen参数的取值必须符合物理学定律和实际地学情况,随意的取值可能导致无意义乃至错误的结果.本文根据热力学定律和弹性常数的物理意义,结合大量的实测数据,提出常见TI介质的Thomsen参数需要满足以下约束条件:1/4f=1-β02/α02ε>-f/2; 1/2f-1δf-1); -1/2γε)/4(1-f)-1/2.Thomsen参数ε、δ、γ的取值区间主要受P、S波参考速度比β0/α0的约束,其值可正可负,实测参数中ε和γ正多负少;δ、γ、β0/α0的取值范围都有上下界,而ε没有上限;γ与ε正相关且取值上限受到ε的限制;ε、γ与δ之间不存在明显的约束.鉴于椭圆各向异性介质和薄互层等效TI介质在实际应用中的普遍性,专门给出了这两种介质各向异性参数的附加约束条件.一些在实际资料中可观测到的P波特殊偏振方向和SV波三叉现象也能为各向异性参数提供额外的约束.之后,我们将TI介质弹性常数和各向异性参数的约束条件扩展到对正交各向异性介质弹性常数和各向异性参数的约束.本文提出的各向异性参数约束条件简单实用,为各向异性理论研究和数值模拟中参数的选择提供依据,为实际资料的各向异性参数反演提供约束,既能避免出现无物理意义的研究结果,又能加速反演的搜索过程,提高生产效率,具有理论指导意义和实际应用价值.

Constraints on anisotropic parameters in transversely isotropic media and the extensions to orthorhombic media

The presence of anisotropy in earth materials has been widely accepted. Due to its high symmetry and relative simplicity in mathematical formulae, transversely isotropic (TI) medium has become one of the most important anisotropic media in seismology. In theoretical and practical studies, anisotropic parameters expressed in terms of elastic constants proposed by Thomsen in 1986 are widely adopted. Values of elastic constants and Thomsen's anisotropic parameters cannot be chosen freely. There are some constraints among those parameters. Unfortunately, such constraints were ignored by some authors. Sometimes improper parameters were selected and consequently physically meaningless and misleading results were generated. In this study, we proposed a set of simple inequalities for constraining the values of Thomsen's anisotropic parameters. The inequalities, given by 1/40²/α₀²<1, ε>-f/2, 1/2f-1<δ<2(1/f-1), and -1/2<γ<(1+2ε)/4(1-f)-1/2, were developed based upon the stability condition of material as well as theoretical and experimental investigations on anisotropic parameters. It was shown that ε, δ and γ can be either positive or negative, and their ranges are mainly constrained by P-to-S velocity ratio β₀/α₀; δ, γ and β₀/α₀ are bounded, while ε has no upper limit; γ and ε are positively correlated, and the upper limit of γ is related to ε; the links between ε and δ, and between γ and δ are weak. It was also proved in this paper that the inequalities about five independent elastic constants of TI media derived from the stability condition are sufficient to guarantee that all the phase velocities of three body waves are non-negative. Constraints for some special TI media and physical phenomena were reviewed. For elliptical anisotropy, ε equals to δ and thus 1/2f-1<ε<2(1/f-1). For equivalent TI medium arising from periodic isotropic thin layers, we have ε≥max(0, δ) and γ≥0. For some directions called longitudinal directions, at which phase velocity of P waves reaches an extremum, the direction of P polarization is coincident with wavenormal and ray direction. The sufficient and necessary condition for the presence of longitudinal direction is given by ε/δ_L<1/2, where δ_L =(C₁₃+2C₅₅-C₃₃)/C₃₃ is the linear definition of Thomsen's δ and C_ij elastic constants. For the presence of SV triplications, the inequality δ-ε/f>(1-f)/2 is the condition for basal triplication, and δ-ε>(1-f)/2 for axial triplication, (ε-δ)/(1+δ)>2(1-f)/3 for oblique triplication. The inequality for oblique triplication is simplified from the inequality proposed by Thomsen and Dellinger (2003). We also extend the constraints on elastic constants and anisotropic parameters of TI media to those of orthorhombic media.