Perturbation Approach to Derive Born Approximation, Frechet Derivative, and Migration Operator Bowen Guo

Goal: to derive s 1.Born modeling equation Migration operator s

Starting point: Helmholtz equation  Specially, when the source is a harmonic point source, equation (1) becomes equation (2) Since, Thus: (1) (2)

Starting point Integrate over both sides (volume integration), Change volume integration to surface integration, Considering Equation (2) we get s V0，S0V0，S0 (3) The first term = 0, if there are only out-going wave and S 0 at infinity

Starting point: inverse operator to the Helmholtz equation Thus, under such assumption Compare with Helmholtz equation We can conclude (4)

Middle point: Lippmann-schwinger equation Assume the velocity is perturbed, Plugging into Helmholtz equation and neglecting the second order perturbation term: Considering equation (4) (5) Green’s function response of the background medium v v v v Background perturbated Change notation

Middle point: 1 st order Born approximation Similarly Only zero and first order term are kept (first order Born approximation). It is assumed that Taylor expansion at x=0 v v x Zero order term = First order term = Second order term = … is small … = …

Final point: 1 st order Born approximation Assuming: We can conclude Source to scatter Scatter to receiver s

Final point: Frechet derivative  For a single point scatter at x 0 background wave field term s s

Final point: matrix notation of Born modeling: d=Lm L m d Summation in model space ! Change notation,,, and sum over all frequency

Final point: a closer watch at d=Lm  d: scattered wave field, which only contains primary reflection and diffraction information, no multiples.  L: forward modeling operator, which requires prior information of G.  m: reflectivity-like model. Velocity Perturbation (reflectivity)

Extension: L T =migration operator  Matrix notation   L T operates on d, giving reflectivity-like image. Assume a 2-dimensional model is then m is a vector N x 1. Assume d is a vector M x 1 then L is a matrix M x N. Usually M > N, the system of equations is over-determined. Usually, because ?

Extension: L T = migration operator (L and L T )  ( giving perturbation m, what is scattered data ?)  (giving scattered data, where are perturbations ?) model space data space

Extension: physical interpretation of migration (using Kirchhoff migration as an example)   For illustration, background velocity is homogeneous. For a fixed s and g pair Where is the scatter? It must be in the place where is satisfied. sg

Extension: different migration methods  Different ways to calculate Green’s function 1.Kirchhoff migration: solve travel times to asymptotically represent Green’s function 2.Reverse time migration: 3.Generalized diffraction stack migration:  All require a good background velocity model (a good G)

Extension: different migration methods  Different ways to calculate Green’s function 1.Kirchhoff migration: solve travel times to asymptotically represent Green’s function 2.Reverse time migration: 3.Generalized diffraction stack migration:  All require a good background velocity model (a good G)

Extension: a re-look at Kirchhoff migration   velocity is heterogeneous ? Eikonal solver to For a fixed s and g pair Where is the scatter? It must be in the place where is satisfied. s g Smearing data in the model where match

Extension: different migration methods  Different ways to implement the following equation: 1.Kirchhoff migration: solve travel times to asymptotically represent Green’s function 2.Reverse time migration: 3.Generalized diffraction stack migration:  All require a good background velocity model (a good G)

Extension: Reverse Time Migration (RTM) Source side Receiver side       A migration image of a single pair of source and receiver Refraction+reflect ion Different combinations make different parts in the image. Smile Rabbit ear Cigar Background velocity

Extension: Reverse Time Migration Refraction Reflection Cigar Rabbit ear r 2 Smile r Rabbit ear r 2 velocity smoothly increase with depth abrupt velocity change n/a Smearing data in the model where match

Extension: L T = migration operator (gradient of L2 misfit function)  Misfit function:  To use a gradient optimization method to minimize the misfit function, the gradient: Since Thus

Summary  Born approximation (modeling):  Frechet derivative (single frequency):  Migration:  Matrix notation:  Different ways to implement migration (Kirchhoff, RTM, GDM)  A closer look at RTM (cigar, rabbit ear, smile)