The acoustic Green’s function in 3D is the impulsive point source response of an acoustic medium. It satisfies the 3-D Helmholtz equation in the frequency domain for an arbitrary and linear acoustic medium with constant density, 2.1 Green’s functions

2.2 Reciprocity equation of convolution type for all x inside the integration volume.

Using the product rule for differentiation (i.e., d(fg) = gdf + fdg), we get:

The integrands in this equation are a product of a monopole source (i.e., point source Green’s function G(x|A)) and a dipole (i.e., dG/dn ≈ [G(x + dn|A) − G(x|A)]/|dn|) source.  the spectral phases are added Adding phases is the same as adding travel-times, suggesting that the monopole-dipole product predicts events with longer traveltimes and raypaths.

If the entire integration surface is at infinity and v(x) = v 0 (x), i.e. G 0 (x|A) = G(x|A), then the integral’s contribution is zero by the Sommerfeld outgoing radiation condition.  the reciprocity property of Green’s functions for an arbitrary velocity distribution: which is also true if the functions are conjugated.

Replacing the terms in the above { } brackets by −δ(x − B) gives The final expression for the field G(B|A) is given by integrating the above equation over the entire volume enclosed by a sphere at infinity: total field scattered field background field Lippmann-Schwinger equation Born forward modeling equation (for weak scatterers, G(x|A)≈ G 0 (x|A) )

2.3 Reciprocity equation of correlation type

Useful Properties 1.Inverse Fourier transforming 2iIm[G(B|A)] = G(B|A) − G(B|A), the causal Green’s function can be recovered by evaluating g(B, t|A, 0) − g(B,−t|A, 0) for t ≥ 0. 2.The reciprocity-correlation equation is characterized by a product of unconjugated and conjugated Green’s functions. The subtraction of phases is the same as subtracting traveltimes, suggesting that the monopole-dipole product predicts events with shorter traveltimes and shorter raypaths.

3. If the medium is sufficiently heterogeneous, the correlation reciprocity integral at infinity vanishes due to strong scattering in the medium. 4. If A is near the free surface and B is along a vertical well then the Green’s functions can be interpreted as either VSP or SSP Green’s functions. The reciprocity equation of correlation type can be rewritten as which is the SSP→VSP transform of the correlation type.

Far-field approximation  product of two monopoles Induced radiation from the scattering body in the far field is where r is the distance between a point B within the scatterer region and the point x far from the scatterer4 and m(A, μ) is a function of the angular coordinates only. The gradient terms in reciprocity equation of correlation type become in the far-field approximation The far-field expression is where the integration along the boundary at infinity can be neglected for a sufficiently heterogeneous medium.

2.4 Stationary phase integration where the integration is over the real line, φ(x) is real and a well-behaved phase function with at most one simple stationary point, ω is the asymptotic frequency variable, and g(x) is a relatively slowly varying function. If the exponential argument ωφ(x) is large and rapidly varying with x then e iωφ(x) is a rapidly oscillating function with a total algebraic area of zero, the line integral goes to zero. The real part of e iωφ(x) for a linear phase function ωφ(x) = kx. Here, k = ω/v is a constant defined as the instantaneous wavenumber d[ω φ(x)]/dx. As ω gets large, the rule is that the oscillation rate k increases and the algebraic area under the curve mostly goes to zero.

An exception at a stationary point x* where the instantaneous wavenumber ωφ(x)′ x=x* = 0. The quadratic term in the Taylor series expansion of φ(x) about x*: This integral is recognized as the Fresnel integral; Substituting this equation into the previous equation yields Substituting this equation into the previous equation yields the asymptotic form where is an asymptotic coefficient.

Let the normalized direct wave G(x|B) = e iωτ xB and the normalized reflected wave G(x|A) = e iωτ Ayox,

2.5 Seismic migration The goal in seismic imaging is to invert the seismic traces for an estimate of the reflectivity distribution. The approximate inverse by applying the adjoint of the forward modeling operator is known as migration or Born inversion. Simply put, seismic migration is the relocation of a trace’s reflection event back to its place of origin, the reflector boundary. Born modeling equation; where D(g|s) represents the shot gather of scattered energy in the frequency domain for a source at s and a geophone at g, and m(x) = 2s(x) δ s(x) approximates the weighted reflectivity distribution. D(g|s)  d ; m(x)  m ; the Born forward modeling operator  L ; The data function d is indexed according to the data coordinates for the source s and receiver g locations, while the model vector m is indexed according to the reflectivity locations x in model space.

Approximated by the matrix-vector equation; where d i and m j represent, respectively, the components of the M×1 data vector d and the N×1 model vector m. Generally, M >> N (overdetermined) and the residual components ≠ 0 for all values of i (inconsistent). To minimize the sum of the squared errors (or the misfit function),

By replacing d and m by their function representations, and plugging this into the previous formula, Compared to the forward modeling equation which sums over the model-space variables x, the migration integral sums over the data space variables s and g; it also includes the extra integration over ω. The kernel is the conjugate of the one in the forward modeling. By replacing the Green’s functions by their asymptotic forms, the diffraction stack formula is obtained.

2.6 Interferometric migration In standard migration; D(g|s) represents the actual scattered data after the direct wave is muted. The Green’s functions G(x|s) and G(x|g) are computed by a model-based procedure such as ray tracing for diffraction stack migration, or by an approximate finite-difference solution to the wave equation. The estimated velocity model always contains inaccuracies that lead to errors in the computed Green’s functions. Such mistakes manifest themselves as defocused migration images. In interferometric migration; The natural data are used to either fully or partly replace the Green’s functions, resulting in no need for the velocity model and avoidance of defocusing errors. Another type is interferometric redatuming followed by a model-based migration of the redatumed traces obtained by first interferometrically redatuming the raw data D(g|s) to a new recording datum closer to the target.