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Reciprocity Relations Reciprocity relations - May 11, 2006 Using the reciprocity theorem a relation between the seismic reflection response and transmission data can be derived. This relation is used to calculate transmission data from reflection data measured at the surface. In this presentation the steps involved in this calculation are explained in detail and possible pitfalls are discussed. At the end a simple example is given to illustrate the discussed procedure. 1 One-way reciprocity One-way reciprocity relation of the convolution type: + − − + 2x + − − + 2x {PA PB − PA PB }d H = {PA PB − PA PB }d H (1) Z∂D0 Z∂Dm One-way reciprocity relation of the correlation type: + ∗ + − ∗ − 2x + ∗ + − ∗ − 2x {(PA ) PB − (PA ) PB }d H = {(PA ) PB − (PA ) PB }d H (2) Z∂D0 Z∂Dm the medium parameters in both states A and B are identical, lossless and 3-D inhomogeneous and the domain D is source free. State A State B x xA x xB z0 ∂D0 ∂D0 R0(x, xA,ω) R0(x, xB ,ω) ∆z ∆z D T0(x, xA,ω) D T0(x, xB ,ω) x zm x ∂Dm ∂Dm 1 Surface ∂D0 Field State A State B + P δ(xH − xH,A)sA(ω) δ(xH − xH,B )sB(ω) − P R0(x, xA,ω)sA(ω) R0(x, xB ,ω)sB(ω) Surface ∂Dm + P T0(x, xA,ω)sA(ω) T0(x, xB ,ω)sB(ω) P − 0 0 One-way reciprocity relation of the convolution type: 2 sAsB {δ(xH − xH,A)R0(x, xB ) − R0(x, xA)δ(xH − xH,B )}d xH =0 (3a) Z∂D0 R0(xA, xB )= R0(xB , xA) (3b) One-way reciprocity relation of the correlation type: x x ∗ x x x x 2x ∗ x x x x 2x δ( H,A − H,B) − R0( , A)R0( , B )d H = T0 ( , A)T0( , B )d H (4) Z∂D0 Z∂Dm Equation (4) can be used to estimate transmission effects on wave propagation (coda) from reflection data at the surface. 2 From Reflection to Transmission Dependency on ω is omitted. x x ∗ x x x x 2x ∗ x x x x 2x δ( H,A − H,B) − R0( , A)R0( , B )d H = T0 ( , A)T0( , B )d H (5) Z∂D0 Z∂Dm 2.1 Forward model of Transmission response The transmission response is written as a flux normalized generalized primary propagator Wg in the ω − p domain: T0(p, xA,zm,z0)= Wg(p, xA,zm,z0) (6) T0(p, xA,zm,z0)= Wp(p, xA,zm,z0)C(p, xA, ∆z) (7) with Wp the primary propagator and C the coda operator and ∆z = zm − z0. The code operator is a causal function describing the distortion of Wg relative to Wp. The effect of multiple reflections can be delaying, shaping and magnifying the pulse transmitted through a layered sequence. 2 xA z0 ∆z C(∆z) Wp(zm,z0) x zm O’Doherty and Anstey [] found that the coda operator C(ω) is related to the power spectrum of the reflection coefficient. See discussion in proefschrift Frederic: pages 18-19-20 section 2.4. For the current discussion we only need the know that C can be written as: C(p, xA, ∆z) = exp(−A(p)∆z) (8) zm 1 −2 2 2 Wp(p, xA,zm,z0) = exp(−jω (c (z) − p ) dz) (9) Zz0 ∗ x x Wp (p, A,zm,z0)Wp(p, A,zm,z0) ≈ 1.0 (10) C(p, xA, ∆z) = exp(−A(p)∆z) (11) ∗ ∗ T (p, xA, ∆z)T (p, xA, ∆z)= C (p, xA, ∆z)C(p, xA, ∆z) (12) ∗ C (p, xA, ∆z)C(p, xA, ∆z) = exp(−2R{A(p)}∆z) (13) R{A}(p) is the real part of A(p). Note that A(p) is written as a function of the ray-parameter p. Replace integral with discrete notation in vectors: ∗ x x x x 2x H x x x x R0( , A)R0( , B )d H ≡ R0 ( , A)R0( , B ) (14) Z∂D0 R0(x0, xA) . . ∗ ∗ ∗ = R (x0, xA) ...R (xi, xA) ...R (xN , xA) R0(xi, xA) 0 0 0 . . R (x , x ) 0 N A extend vectors to matrices by including more shot records (different xA) Matrices for propagation and 3 reflection including all effects due to interbed multiples with a fixed spread acquisition geometry: C(xr,0, xs,0) ... C(xr,0, xs,N ) . C = . .. (15a) C(x , x ) ... C(x , x ) r,M s,0 r,M s,N R(xr,0, xs,0) ... R(xr,0, xs,N ) . R = . .. (15b) R(x , x ) ... R(x , x ) r,M s,0 r,M s,N represents a fixed spread acquisition geometry. Using the matrices the integral equations becomes: TH T = I − RH R (16) H H H T T = (WpC) WpC = C C (17) CH C = I − RH R (18) C is measured at ∂Dm and R is measured at ∂D0. Superscript H denotes complex-conjugatetranspose. For fixed spread geometry both R and T matrices are symmetric (not Hermitian) due to the reciprocity relation between source and receiver positions. We need to resolve C from CH C and make the assumption that: H C = LΛcL (19) Λc(p) = exp {−A(p)(ω)∆ztot} (20) e−A(p1)∆ztot 0 . 0 e0 e−A(p2)∆ztot . 0 Λ (p)= (21) c ............................................. 0 0 ... e−A(pN )∆ztot where A(p) is a diagonalmatrix. For plane waves in 1D media C is a circulant matrix which has the property Λ F that its Fourier transform is equal to its eigenvalues (see [Sj¨ostr¨om, 1996] for proof): c = x→kx {C} C = FH Λ F e c Correlation of coda matrix: H ΛH Λ H C C = L c cL (22) ΛH Λ R c c = exp {−2 {A(p)}∆ztot} (23) e−2R{A1} 0 . 0 −2R{A2} H 0 e e . 0 Λ Λ = (24) c c ..................................... 0 0 ... e−2R{AN } Using the assumptions about the coda operator the A(p) operator can be retrieved from its real part. The main assumption states that the elements Al(ω) of A(p) are the Fourier transforms of causal filters in the time domain. This is true for 1D media, since in that case A is the spatial fourier transform of C, so e A˜ Al(ω)= l(kx,l,ω). Note that before the temporal Fouriere transforms yields the causal filters, the diagonal e e 4 1 elements must to be scaled from wavenumber kx to propagation angle p with ω (kx = pω). Also in the Λ 1 general 3-D case a scaling of the diagonal of c with ω is reguired before the Al(ω) can be interpreted as the Fourier transforms of causal filters in the time domain. Now R{Ai} can be calculated using: e R A H 2 { i} = ln(Λc,iΛc,i) (25) H Since in general Λc,iΛc,i is a complex number a complex logarithm ln(a + ib) must be taken: 1 R{ln(a + ib)} = (a2 + b2) 2 (26) b I{ln(a + ib)} = atan( ) (27) a Reconstruction of Ai(p) from its real part is possible because Ai(p) is a causal function and with the definition of the analytic signal: 1 +∞ R{A(ω′)} R ′ (28) A(ω)= {A(ω)} + ′ dω jπ Z−∞ ω − ω it can be reconstructed. In practice A(p,ω) is transformed back to time with a complex to complex Fourier transform ar(p,t)= Fω→t{A(p,ω)} and then the real part is taken. Proof see [Bracewell, 1986]: A real signal b(t) has a causal impulse response of a(t)= H(t)b(t) (29a) a(t)= O(t)+ E(t)(splitinevenandoddsignal) (29b) 1 1 a(t)= [a(t)+ a(−t)] + [a(t) − A(−t)] (29c) 2 2 O(t)= sign(t)E(t) (29d) a(t)=(1+ sign(t))E(t) (29e) −2 A(ω)= G(ω)+ i( ) ∗ G(ω) (29f) ω where H(t) is the Heaviside step function: 0 t< 0 1 H(t)k = 2 t =0 (30) 1 t> 0 2.2 Inverse model of Transmission response Correlation of reflection matrix: H H R R = LΛrL (31) 5 xA xB → x z0 ∆z R(x, xA,zm,z0) R(x, xB ,zm,z0) zm Flow diagram of computational tasks: H H R(x,ω) R R LΛrL Λr(p) A(p,ω) exp {−A(p,ω)} T(p,ω) Details for different media: ∗ ∗ R (xr,0, xs,0) ... R (xr,M , xs,0) R(xr,0, xs,0) ... R(xr,0, xs,N ) H . R R = . .. . .. (32a) R∗(x , x ) ... R∗(x , x ) R(x , x ) ... R(x , x ) r,0 s,N r,M s,N r,M s,0 r,M s,N For a 1 dimensional medium this reduces to: R∗R(0) ... R∗R(N) H . R R = . .. (32b) R∗R(N) ... R∗R(0) where R∗R is the auto-correlation function of the reflection response R. In the wavenumber domain RH R can be easily calculated using the Fourier transform of one shot record and compute the autocorralation in the wavenumber domain (in wavenumber domain all elements of the computation l = re2 + im2 are real).[Efficient algorithm for eigenvalue decomposition of Toeplitz matrix...., with circulant matrix as pre- conditioner.] The resulting cross-correlation matrix RH R in the spatial domain for 1D media is a Hermitian Toeplitz matrix (which has real eigenvalues, add proof for Hermitian matrix has real eigenvalues). Check that RH R has a diagonal with real elements.
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