Shear wave splitting

Shear wave splitting, also called seismic birefringence, is the phenomenon that occurs when a polarized shear wave enters an anisotropic medium (Fig. 1). The incident shear wave splits into two polarized shear waves (Fig. 2). Shear wave splitting is typically used as a tool for testing the anisotropy of an area of interest. These measurements reflect the degree of anisotropy and lead to a better understanding of the area's crack density and orientation or crystal alignment.We can think of the anisotropy of a particular area as a black box and the shear wave splitting measurements as a way of looking at what is in the box. ∂ ∂ x i [ c i j k l ∂ U k ∂ x l ] = ρ ∂ 2 U j ∂ t 2 {displaystyle {frac {partial }{partial x_{i}}}left= ho {frac {partial ^{2}U_{j}}{partial t^{2}}}}     (1) t = τ ( x i ) {displaystyle t= au left(x_{i} ight)}     (2) U k ( x i , t ) = ∑ n = 0 ∞ U k ( n ) ( x i ) f n ( t − τ ( x i ) ) {displaystyle U_{k}left(x_{i},t ight)=sum _{n=0}^{infty }U_{k}^{left(n ight)}left(x_{i} ight)f_{n}left(t- au left(x_{i} ight) ight)}     (3) d f n + 1 ( ϑ ) / d ϑ = f n ( ϑ ) {displaystyle df_{n+1}left(vartheta ight)/dvartheta =f_{n}left(vartheta ight)}     (4) N ( U ( n ) ) − M ( U ( n − 1 ) ) + L ( U ( n − 2 ) ) = 0 {displaystyle Nleft(U^{left(n ight)} ight)-Mleft(U^{left({n-1} ight)} ight)+Lleft(U^{left({n-2} ight)} ight)=0}     (5) { N j ( U ( n ) ) = Γ j k U k ( n ) − U j ( n ) M j ( U ( n ) ) = p i   a i j k l ∂ U k ( n ) ∂ x l + ρ − 1 ∂ ∂ x i ( ρ   a i j k l   p l U k ( n ) ) L j ( U ( n ) ) = ρ − 1 ∂ ∂ x i ( ρ   a i j k l ∂ U k ( n ) ∂ x l ) {displaystyle {egin{cases}N_{j}left(U^{left(n ight)} ight)=Gamma _{jk}U_{k}^{left(n ight)}-U_{j}^{left(n ight)}\M_{j}left(U^{left(n ight)} ight)=p_{i}~a_{ijkl}{frac {partial U_{k}^{left(n ight)}}{partial x_{l}}}+ ho ^{-1}{frac {partial }{partial x_{i}}}left( ho ~a_{ijkl}~p_{l}U_{k}^{left(n ight)} ight)\L_{j}left(U^{left(n ight)} ight)= ho ^{-1}{frac {partial }{partial x_{i}}}left( ho ~a_{ijkl}{frac {partial U_{k}^{left(n ight)}}{partial x_{l}}} ight)end{cases}}}     (6) Γ j k = p i   p l   a i j k l , a i j k l = c i j k l / ρ , p i = ∂ τ ∂ x i {displaystyle Gamma _{jk}=p_{i}~p_{l}~a_{ijkl},quad a_{ijkl}=c_{ijkl}/ ho ,quad p_{i}={frac {partial au }{partial x_{i}}}}     (7) N j ( U ( 0 ) ) = Γ j k U k ( 0 ) − U j ( 0 ) == ( Γ j k − δ j k ) U k ( 0 ) == 0 {displaystyle N_{j}left(U^{left(0 ight)} ight)=Gamma _{jk}U_{k}^{left(0 ight)}-U_{j}^{left(0 ight)}==left(Gamma _{jk}-delta _{jk} ight)U_{k}^{left(0 ight)}==0}     (8) D e t ( Γ j k − G δ j k ) = 0 {displaystyle Detleft(Gamma _{jk}-Gdelta _{jk} ight)=0}     (9) G 3 − P G 2 + Q G − R = 0 {displaystyle G^{3}-PG^{2}+QG-R=0}     (9) u ( ω ) = A   w ( ω ) exp ⁡ [ − i ω T 0 ] ⋅ p ^ {displaystyle uleft(omega ight)=A~wleft(omega ight)exp leftcdot {hat {p}}}     (10) Γ = exp ⁡ [ i ω δ t / 2 ] f ^ f ^ + exp ⁡ [ − i ω δ t / 2 ] s ^ s ^ {displaystyle Gamma =exp left{hat {f}}{hat {f}}+exp left{hat {s}}{hat {s}}}     (11) u s ( ω ) = A   w ( ω ) exp ⁡ [ − i ω T 0 ] Γ ( ϕ , δ t ) ⋅ p ^ {displaystyle u_{s}left(omega ight)=A~wleft(omega ight)exp leftGamma left(phi ,delta t ight)cdot {hat {p}}}     (12) Shear wave splitting, also called seismic birefringence, is the phenomenon that occurs when a polarized shear wave enters an anisotropic medium (Fig. 1). The incident shear wave splits into two polarized shear waves (Fig. 2). Shear wave splitting is typically used as a tool for testing the anisotropy of an area of interest. These measurements reflect the degree of anisotropy and lead to a better understanding of the area's crack density and orientation or crystal alignment.We can think of the anisotropy of a particular area as a black box and the shear wave splitting measurements as a way of looking at what is in the box. An incident shear wave may enter an anisotropic medium from an isotropic media by encountering a change in the preferred orientation or character of the medium. When a polarized shear wave enters a new, anisotropic medium, it splits into two shear waves (Fig.2).One of these shear waves will be faster than the other and oriented parallel to the cracks or crystals in the medium. The second wave will be slower than the first and sometimes orthogonal to both the first shear wave and the cracks or crystals in the media. The time delays observed between the slow and fast shear waves give information about the density of cracks in the medium. The orientation of the fast shear wave records the direction of the cracks in the medium.