GEOPHYSICS, VOL. 75, NO. 6 共NOVEMBER-DECEMBER 2010兲; P. S225–S235, 7 FIGS. 10.1190/1.3506146

Derivation of forward and adjoint operators for least-squares shot-profile split-step migration

Sam T. Kaplan1, Partha S. Routh2, and Mauricio D. Sacchi1

In the context of least-squares migration, these are, respectively, the adjoint and forward operators for the least-squares normal equations. The algorithm that solves this set of normal equations, we call shot-profile least-squares migration. Least-squares migration has received ample attention in the geophysical literature. Keys and Weglein 共1983兲 introduced a generalized linear inversion for the Born approximation, allowing for incomplete data and prior information. Their paper presents the first nondirect 共i.e., least-squares兲 solver for the Born approximation. Concurrently, authors discussed the Born approximation and its direct inverse solution using constant velocity Green’s functions 共Cohen and Bleistein, 1979; Raz, 1981; Weglein and Gray, 1983兲 and WKBJ Green’s functions 共e.g., Clayton and Stolt, 1981; Foster and Carrion, 1984; Bleistein and Gray, 1985兲. On a parallel track, a succession of wavefield propagation techniques for migration developed, for example, allowing for migration velocity models that vary in depth only 共Gazdag, 1978兲.Amultitude of approximations, allowing for laterally varying velocity models followed 共e.g., Gazdag and Sguazzero, 1984; Stoffa et al., 1990; Wenzel, 1991; Kessinger, 1992; Popovici, 1996兲. With the privilege of hindsight, one may choose to take the perspective that these wavefield propagators are approximations to the Green’s function in the Born approximation 共e.g., Huang et al., 1999兲. This is the perspective that we take in this paper. In the above discussion, we have neglected several methods, notably those using finite difference 共Claerbout, 1970兲, as well as asymptotic methods. In particular, due to its relevance to the development of least-squares migration algorithms, we note that Kirchhoff migration, an integral formulation introduced to geophysics by Schneider 共1978兲 predates many of the referenced wave-equationbased methods. In addition to the choice of propagator, one must choose a geometry for the seismic experiment, or equivalently, the domain in which the migration algorithm operates. The simplest, computationally, of these is coincident source and receiver 共zero-offset migration兲. With the advent of larger computers, prestack migration has increased in popularity. Within the regime of prestack migration, there are two geometries commonly used: source-receiver and shot-profile. Al-

ABSTRACT The forward and adjoint operators for shot-profile leastsquares migration are derived. The forward operator is demigration, and the adjoint operator is migration. The demigration operator is derived from the Born approximation. The process begins with a Green’s function that allows for a laterally varying migration velocity model using the split-step approximation. Next, the earth is divided into horizontal layers, and within each layer the migration velocity model is made to be constant with respect to depth. For a given layer, 共1兲 the source-side wavefield is propagated down to its top using the background wavefield. This gives a background wavefield incident at the layer’s upper boundary. 共2兲 The layer’s contribution to the scattered wavefield is computed using the Born approximation to the scattered wavefield and the background wavefield. 共3兲 Next, its scattered wavefield is propagated back up to the measurement surface using, again, the background wavefield. The measured wavefield is approximated by the sum of scattered wavefields from each layer. In the derivation of the measured wavefield, the shot-profile migration geometry is used. For each shot, the resulting wavefield modeling operator takes the form of a Fredholm integral equation of the first kind, and this is used to write down its adjoint, the shot-profile migration operator. This forward/adjoint pair is used for shot-profile least-squares migration. Shot-profile least-squares migration is illustrated with two synthetic examples. The first uses data collected over a four-layer acoustic model, and the second uses data from the Sigsbee 2a model.

INTRODUCTION We derive and implement operators for shot-profile migration and demigration using wavefield propagators from split-step migration.

Manuscript received by the Editor 12 November 2009; revised manuscript received 21 May 2010; published online 2 December 2010. 1 Department of Physics, University of Alberta, Edmonton, Alberta, Canada. E-mail: [email protected]; [email protected]. 2 Upstream Research Company, ExxonMobil, Houston, Texas, U.S.A. E-mail: [email protected], [email protected]. © 2010 Society of Exploration Geophysicists. All rights reserved.

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though the equivalence between the two geometries has been shown 共Wapenaar and Berkhout, 1987; Biondi, 2003兲, they differ in their respective parameterizations of the prestack migrated image gathers. From a practitioner’s point of view, this difference in parameterization is important 共Jeannot, 1988兲. The first widely cited practical implementation of the ideas introduced by Keys and Weglein 共1983兲 is Nemeth et al. 共1999兲 and uses Kirchhoff operators to propagate the wavefield. Later, Kühl and Sacchi 共2003兲 and Clapp 共2005兲 used wave-equation operators as well as introducing angle versus ray-parameter image gathers to leastsquares migration. In both cases, the algorithms can be classified as type source-receiver. These implementations validate the ideas in Keys and Weglein 共1983兲 that, for example, generalized inversion can compensate for incomplete data. Other relevant examples of least-squares migration include Ji 共1997兲, Chavent and Plessix 共1999兲, Duquet et al. 共2000兲, Wang et al. 共2005兲, Valenciano et al. 共2006兲, and Symes 共2008兲. We note that Chavent and Plessix 共1999兲, Valenciano et al. 共2006兲, and Symes 共2008兲 work to produce efficient algorithms by using an approximation to the inverse Hessian matrix. We derive demigration 共forward兲 and migration 共adjoint兲 operators for least-squares migration using the shot-profile geometry, the Born approximation, and the split-step approximation. Previously, Rickett 共2001, 2003兲 described similar algorithms for demigration and migration using operator notation. In this paper, we write down, explicitly, the algebra and use the operators in a least-squares migration problem, showing results from two examples. The first uses a four-layer acoustic model, and corrupts the data with random noise and dead traces. The second example uses a single-shot gather from the Sigsbee 2a model. The contribution of this paper is the formal derivation of forward and adjoint operators for split-step waveequation shot-profile migration and their application to least-squares migration. In our derivations, we choose to approximate the Green’s function using the split-step method. Alternative derivations could be built using more comprehensive approximations to the Green’s function, using, for example, the extended split-step 共Kessinger, 1992兲 or phase shift plus interpolation 共Gazdag and Sguazzero, 1984兲 methods.

SHOT-PROFILE WAVEFIELD MODELING In the context of least-squares migration, the forward operator is wavefield modeling 共often referred to as de-migration兲. We outline the construction of the forward operator using 共1兲 the Born approximation, 共2兲 a constant velocity Green’s function altered for laterally varying velocity using the split-step approximation, and 共3兲 a Gazdag depth marching algorithm for depth-varying velocity. For the Born approximation to the wavefield, we write ␺ ⬇ ␺ d Ⳮ ␺ s, where ␺ d is the direct 共i.e., background兲 wavefield,

Figure 1. We give a schematic description of the first two terms in wavefield modeling, ␺ s共1兲 and ␺ s共2兲.

␺ d共xg,zg兩xs,zs;␻ 兲 ⳱ f共␻ 兲G0共xg,zg兩xs,zs;␻ 兲,

共1兲

and the Born approximation for the scattered wavefield is ␺ s,

␺ s共xg,zg兩xs,zs;␻ 兲 ⳱ f共␻ 兲 ⫻



冕冕



ⳮ⬁

G0共xg,zg兩x⬘,z⬘;␻ 兲

␻ c0共x⬘,z⬘兲



2

␣ 共x⬘,z⬘兲

⫻G0共x⬘,z⬘兩xs,zs;␻ 兲dx⬘dz⬘,

共2兲

c20共x⬘,z⬘兲 , c2共x⬘,z⬘兲

共3兲

where,

␣ 共x⬘,z⬘兲 ⳱ 1 ⳮ

is called the scattering potential 共e.g., Weglein et al., 2003兲. In equations 1–3, xg ⳱ 共xg,y g兲 and xs ⳱ 共xs,y s兲 are, respectively, lateral geophone and source positions, with zg and zs being their respective depths below the surface. The function f共 ␻ 兲 is the frequency distribution of the seismic source, and G0 is a Green’s function. Within the context of least-squares migration, ␣ 共with a chosen parameterization兲 is the migrated image, c is the earth’s velocity, and c0 is the migration velocity. We have assumed the scalar wave-equation. To evaluate equations 1 and 2 when the migration velocity c0 varies only in its lateral dimension 共that is, c0 ⳱ c0共x兲兲, we use a constant velocity Green’s function 共e.g., DeSanto, 1992兲 in conjunction with the split-step approximation 共Stoffa et al., 1990兲, giving, ⳮ1

G0共x,z兩x⬘,z⬘;␻ 兲 ⳱ ei␻ 共c0 ⫻

共x兲ⳮcⳮ1 1 兲兩zⳮz⬘兩

冕冉 ⬁

ⳮ⬁



1 i4kz共c1兲

冉 冊 冊 1 2␲

2

⫻eⳮikx·共x⬘ⳮx兲eikz共c1兲兩zⳮz⬘兩dkx,

共4兲

where the split-step approximation is used to allow the vertical wave-number to depend on the lateral dimensions x. In particular, it uses an approximation involving a truncated Taylor expansion and resulting in, ⳮ1 kz共x兲 ⬇ kz共c1兲 Ⳮ ␻ 共cⳮ1 0 共x兲 ⳮ c1 兲.

共5兲

In equations 4 and 5, z is depth, x ⳱ 共x,y兲 are lateral space, kx ⳱ 共kx,ky兲 are lateral wave-number, cⳮ1 is the mean of cⳮ1 1 0 共x兲, and kz共c1兲 is the vertical wave-number given by the dispersion relation,

kz共c1兲 ⳱ ⳮsgn共␻ 兲冑␻ 2 /c21 ⳮ kx · kx .

共6兲

Here, in equation 6, we use the split-step approximation to accommodate a laterally varying migration velocity model. Other, more accurate, but computationally expensive, approximations include extended split-step 共Kessinger, 1992兲 and phase shift plus interpolation 共Gazdag and Sguazzero, 1984兲. To allow the migration velocity model c0 to vary in both x and z, we employ, in conjunction with equation 4, the wavefield propagator described in Gazdag 共1978兲. What follows is our description of shotprofile split-step wavefield modeling 共demigration兲. Figure 1 illustrates the propagator for the de-migration operator. In equation 7, the domain of the reference velocity c0共x,z兲 is partitioned into nz domains 共layers兲,

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L.S. shot-profile split-step migration

Dᐉ ⳱ 兵共x,z兲 苸 R3兩0 ⱕ zᐉⳮ1 ⱕ z ⬍ zᐉ其,

␺ s ⳱ ␺ s共1兲 Ⳮ ␺ s共2兲 Ⳮ ¯ Ⳮ ␺ s共nz兲 .

冉 冊冉 冊冕 ␻

2

c1共1兲

1 2␲

4

z1

1兲

2兲

␺ s共1兲共xg,z0兩xs,z0;␻ 兲

us共1兲F*u p共1兲F

z0

⫻关us共1兲F*u p共1兲g共kgx,␻ ;xs兲兴␣ 共xg,z⬘兲dz⬘ .

共9兲

In equation 9, F denotes the unnormalized 2D Fourier transform over lateral coordinates xg, and F* its adjoint operation. In equation 10, the function u p共1兲 is,

共12兲

and continue using this simplification 共equation 12兲 in the remaining derivations. The construction of ␺ s共2兲 is illustrated in the second term of Figure 1, and depicts the following three steps used in its computation.

共8兲

We proceed to describe ␺ s共1兲 and ␺ s共2兲, contributions to the scattered wavefield from the scattering potential housed within, respectively, D1 and D2. We use these to infer a general solution for ␺ s共ᐉ兲, the contribution to the scattered wavefield from Dᐉ. In our derivation, we choose parameterizations that force a shot-profile de-migration algorithm, rather than, for example, a source-receiver algorithm. In the first layer D1, we define a measurement surface z0 so that zg ⳱ zs ⳱ z0, and substitute G0共1兲 into equation 2 finding,



共␻ /c0共ᐉ兲共x⬘兲兲2␣ 共x⬘,z⬘兲 ⬇ 共␻ /c1共ᐉ兲兲2␣ 共x⬘,z⬘兲,

ᐉ ⳱ 1 . . . nz, 共7兲

and within each layer there is a corresponding Green’s function G0共ᐉ兲 共equation 4兲 built using its constant 共with respect to depth兲 migration wave-speed c0共ᐉ兲共x兲 that, in turn, approximates c0共x,z兲 for zᐉⳮ1 ⱕ z ⳮ1 ⬍ zᐉ. To accommodate the split-step approximation in Dᐉ, we let c1共ᐉ兲 ⳮ1 be the mean of c0共ᐉ兲共x兲. Each layer contributes to the total scattered wavefield ␺ s such that the contribution from the ᐉth layer is denoted ␺ s共ᐉ兲. Then, by super-position the total scattered wavefield is seen in equation 8,

S227

3兲

We consider the wavefield in D1. We compute the background wavefield 共see equation 1兲 at z1 due to the source at z0. This represents the direct propagation, using G0共1兲, of the wavefield from the point source at z0 to the bottom of D1 共top of D2兲.We denote this wavefield as ␺ d共1兲共x⬘,z1 兩 xs,z0;␻ 兲. We consider the wavefield in D2. We compute the scattered wavefield using the scattering potential within D2, and the boundary condition at the top of D2 given by ␺ d共1兲共x⬘,z1 兩 xs,z0;␻ 兲. Here, the energy is propagated according to G0共2兲 and equation 2, and we denote the resulting wavefield as ␺ s共2,1兲共x⬘,z1 兩 x⬙,z0;␻ 兲. We again consider the wavefield in D1. This time, we compute the background wavefield at z0 due to the boundary condition at the bottom of D1 given by ␺ s共2,1兲共x⬘,z1 兩 x⬙,z0;␻ 兲. Here, the energy is propagated according to G0共1兲, and the resulting wavefield is ␺ s共2兲共xg,z0 兩 xs,z0;␻ 兲.

In the above three-step procedure, we have assumed no reflected energy at the boundaries of Dᐉ. Indeed, this is the single scattering approximation often made in migration. The sum total of the above three steps gives the following contribution 共equation 13兲 to the scattered wavefield from the second layer 共see Appendix A兲,

␺ s共2兲共xg,z0兩xs,z0;␻ 兲 ⳱

冉 冊冉 冊冕 ␻

c1共2兲

2

1 2␲

8

z2

us共1兲

z1

⫻F*u p共1兲Fus共2兲F*u p共2兲F

eikgz共ᐉ兲共z⬘ⳮzᐉⳮ1兲 u p共ᐉ兲共kgx,z⬘;␻ 兲 ⳱ ⳮ , i4kgz共ᐉ兲

共10兲

⫻ 关us共2兲F*u p共2兲Fus共1兲F*u p共1兲g共kgx,␻ ;xs兲兴␣ 共xg,z⬘兲dz⬘, 共13兲

And in equation 11 us共1兲 is,

us共ᐉ兲共xg,z⬘;␻ 兲 ⳱ e

ⳮ1 ⳮ1 i␻ 共c0共ᐉ兲 共xg兲ⳮc1共ᐉ兲 兲共z⬘ⳮzᐉⳮ1兲

,

共11兲

␺ s共ᐉ兲共xg,z0兩xs,z0;␻ 兲

both for ᐉ ⳱ 1. The function,

g共kgx,␻ ;xs兲 ⳱ f共␻ 兲e

where u p共2兲 is given by equation 10 with ᐉ ⳱ 2. Generalizing to the ᐉth layer Dᐉ, we find,

ⳮikgx·xs

,

is the synthetic source used in shot-profile migration algorithms, and kgz共ᐉ兲 is short-hand for kgz共c1共ᐉ兲兲 given by the dispersion relation in equation 6. Equations 10 and 11 are constructed from G0共ᐉ兲, governing the propagation of energy in Dᐉ. The detailed relation between G0共ᐉ兲, u p共ᐉ兲, us共ᐉ兲, and g are presented in Appendix A. In short, in equation 9, a Green’s function propagates the energy from the point source to all potential scattering points, and a second Green’s function propagates energy from the scattering points back to the measurement surface. This explanation is also illustrated by the first term in Figure 1. In addition, we make a simplifying approximation for one of the amplitude terms in equation 2. Namely, we let



冉 冊冉 冊 冕 ␻

c1共ᐉ兲

2

1 2␲

4ᐉ

zᐉ

us共1兲F*u p共1兲Fus共2兲

zᐉⳮ1

⫻F*u p共2兲F ¯ us共ᐉ兲F*u p共ᐉ兲F关us共ᐉ兲F*u p共ᐉ兲Fus共ᐉⳮ1兲 ⫻F*u p共ᐉⳮ1兲F ¯ us共1兲F*u p共1兲g共kgx,␻ ;xs兲兴␣ 共xg,z⬘兲dz⬘, 共14兲 where, again, u p共ᐉ兲 and us共ᐉ兲 are given by equations 10 and 11. The wavefield modeling operator is, then, defined by equations 8 and 14. To develop an efficient implementation, we must analyze the sum in equation 8. We begin by adding ␺ s共1兲 and ␺ s共2兲, approximating the integrals with Riemann sums, so that,

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␣ †共xg,zᐉ;xs兲 ⳱ ⌬z 兺 u*共␻ j,zᐉ;xg,xs兲␺ s共xg,z0兩xs,z0;␻ j兲 共20兲

␺ s共1兲 Ⳮ ␺ s共2兲 ⳱ ⌬zus共1兲F*u p共1兲F兵共2␲ 兲ⳮ4共␻ /c1共1兲兲2

j

⫻关us共1兲F*u p共1兲g共kgx,␻ ;xs兲兴

where, as before, 共 · 兲* denotes the adjoint, so that,

⫻␣ 共xg,z1兲 Ⳮ 共2␲ 兲ⳮ8共␻ /c1共2兲兲2us共2兲F*u p共2兲 ⫻F关us共2兲F*u p共2兲Fus共1兲F*u p共1兲g共kgx,␻ ;xs兲兴␣ 共xg,z2兲其,

* F *u * F ¯ u * u*共␻ j,zᐉ;xg,xs兲 ⳱ 共2␲ 兲ⳮ4ᐉ共␻ j /c1共ᐉ兲兲2关us共ᐉ兲 p共ᐉ兲 s共1兲 * ⫻F*u*p共1兲g*共kgx,␻ j;xs兲兴us共ᐉ兲

共15兲 where we assume a constant thickness ⌬z for each layer Dᐉ. In equation 15, we arranged terms such that an iteration and a recursion can be recognized. In particular, generalizing to nz layers, we recognize in equation 15 an iterative method for downward continuing the source side wavefield into the earth,

vs共1兲共xg,␻ ;xs兲 ⳱ us共1兲F*u p共1兲g共kgx,␻ ;xs兲 vs共ᐉ兲共xg,␻ ;xs兲 ⳱ us共ᐉ兲F*u p共ᐉ兲Fvs共ᐉⳮ1兲,

ᐉ ⳱ 2 . . . nz, 共16兲

and we recognize a recursion 共ᐉ ⳱ nz . . . 1兲 for constructing the wavefield at the measurement surface,

␺ s共xg,z0兩xs,z0;␻ 兲 ⳱ ⌬zv1

* F*u* F, ⫻F*u*p共ᐉ兲F ¯ us共1兲 p共1兲

and we have used,

共F*u p共ᐉ兲F兲* ⳱ F*u*p共ᐉ兲F. In the adjoint 共equations 20 and 21兲, we recognize the well-known split-step shot-profile migration algorithm. In particular, there are two iterations for the downward continuation of the source and receiver side wavefields, as well as the imaging condition 共the sum over frequency indices j in equation 20兲. In particular, the iteration that downward continues the source side wavefield from depth index ᐉ ⳮ 1 to depth index ᐉ is, * 共x ,␻ ;x 兲 ⳱ u* F*u* g*共k ,␻ ;x 兲 vs共1兲 gx j s g j s s共1兲 p共1兲

vᐉ共xg,␻ ;xs兲 ⳱ us共ᐉ兲F*u p共ᐉ兲F共vs共ᐉ兲

⫻共2␲ 兲ⳮ4ᐉ共␻ /c1共ᐉ兲兲2␣ 共xg,zᐉ兲 Ⳮ vᐉⳭ1兲, 共17兲 where vnzⳭ1 ⳱ 0. Hence, as noted in Rickett 共2001兲, the algorithm requires two passes through depth, first to compute and store vs共ᐉ兲 in equation 16, and second to compute the recursion in equation 17. Equations 16 and 17 constitute an algorithm for wavefield modeling which is, in other words, the forward operator for least-squares shot-profile migration. Because we have parameterized the equations for the shot-profile geometry rather than the source-receiver geometry, we can model each shot gather independently.

* 共x ,␻ ;x 兲 ⳱ u* F*u* Fv* , vs共ᐉ兲 g j s s共ᐉ兲 p共ᐉ兲 s共ᐉⳮ1兲

共18兲

where ␻ j is some realization of ␻ , and,

u共␻ j,zᐉ;xg,xs兲 ⳱ 共2␲ 兲ⳮ4ᐉ共␻ j /c1共ᐉ兲兲2us共1兲F*u p共1兲F ¯ us共ᐉ兲

* 共x ,␻ ;x 兲 ⳱ u* F*u* Fv* , vr共ᐉ兲 g j s s共ᐉ兲 p共ᐉ兲 r共ᐉⳮ1兲

ᐉ ⳱ 2 . . . n z,

␣ †共xg,zᐉ;xs兲 ⳱ ⌬z 兺 j

冉 冊冉 冊 1 2␲

4ᐉ



2

c1共ᐉ兲

共24兲

Equations 20 and 21 constitute split-step shot-profile migration, and for efficiency is implemented using the iterations 共downward continuations兲 in equations 22 and 23, and the imaging condition in equation 24.

SHOT-PROFILE LEAST-SQUARES MIGRATION We let d be a vector of length M realized from ␺ s共xg,z0 兩 xs,z0;␻ 兲, and where,

M ⳱ n sn ␻ n g, ns are the number of shot gathers, ng are the number of geophones in each shot gather, and n␻ are the number of frequencies. Likewise, we let m be a vector of length N realized from, ns

␣ 共xg,z兲 ⳱ 兺 ␣ 共xg,z;xs共ᐉ兲兲, ᐉ⳱1

共19兲 where,

With de-migration cast into the form of equations 18 and 19, we write down their adjoint 共e.g., Hansen, 1998兲,

共23兲

so that equation 20 becomes,

⫻ F*u p共ᐉ兲F关us共ᐉ兲F*u p共ᐉ兲F ¯ us共1兲 ⫻ F*u p共1兲g共kgx,␻ j;xs兲兴.

共22兲

* 共x ,␻ ;x 兲 ⳱ u* F*u* F␺ 共x ,z 兩x ,z ;␻ 兲 vr共1兲 g j s s g 0 s 0 s共1兲 p共1兲

* 共x ,␻ ;x 兲v* 共x ,␻ ;x 兲. ⫻vs共ᐉ兲 g j s r共ᐉ兲 g j s

Equations 8 and 14 are wave-equation shot-profile de-migration using split-step operators. For least-squares shot-profile migration, they are the forward operator. In this section, we compute the corresponding adjoint operator, enabling a least-squares formulation. To proceed, we rewrite the forward operator 共equations 8 and 14兲 so that a discretized Fredholm integral equation of the first kind is recognized. In particular, we have for lateral shot location xs,



ᐉ ⳱ 2 . . . nz .

Second, the downward continuation of the receiver side wavefield is,

SHOT-PROFILE WAVEFIELD MIGRATION

␺ s共xg,z0兩xs,z0;␻ j兲 ⳱ ⌬z 兺 u共␻ j,zᐉ;xg,xs兲␣ 共xg,zᐉ;xs兲,

共21兲

N ⳱ n gn z,

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L.S. shot-profile split-step migration

model-norm is two-fold. First, it decreases the condition number of the matrix that is inverted,

and, as before, nz are the number of depths in the migration image gathers. Then we let Aᐉ be the M ⫻ N matrix built from the shot-profile wavefield modeling operator defined by equations 16 and 17, and AHᐉ its adjoint 共migration operator兲 defined by equations 20 and 21, all for the ᐉth shot gather. Next, in equation 25, we define a matrix A such that,

AH ⳱ 关AH 1

AH 2

¯

AnH 兴.

AHWHWA Ⳮ ␮I. This means that it forces a unique solution when otherwise an infinite number of solutions would serve equally well to minimize the data-misfit term in equation 26. Second, it gives some prior notion of the character of the solution. From a Bayesian perspective, our chosen model-norm follows from a Gaussian distribution for the model, with a zero-mean and some prescribed variance 共e.g., Ulrych et al., 2001兲. The trade-off parameter ␮ proportions the importance of fitting the data and honoring the model-norm. If, again, one takes a Bayesian perspective, then the trade-off parameter can be written as ␮ ⳱ ␴ 2n / ␴ m2 , where ␴ n is the standard deviation of noise in the data, and ␴ m is the standard deviation of the model. Therefore, one could estimate ␮ by using an estimate of noise in the data, and some further estimate of the energy in the model. This is the approach that we take in this article. One could use other traditional methods from inverse theory to estimate ␮ such as L-curve analysis 共e.g., Hansen, 1992兲, chi-squared statistics 共e.g., LaBrecque et al., 1996兲, and cross-validation 共e.g., Wahba, 1977兲, but at the expense of solving equation 27 for multiple instances of the trade-off parameter ␮.

共25兲

s

The matrix A maps from m to d, and to find an optimal migrated image, we let dobs be observed data 共common shot gathers兲, and minimize the cost function,

␾ 共m兲 ⳱ 储W共Am ⳮ dobs兲储22 Ⳮ ␮储m储22 .

共26兲

Finding the minimum of equation 26, gives the set of least-squares normal equations,

共AHWHWA Ⳮ ␮I兲m ⳱ AHWHWdobs,

共27兲

which we solve for m. When m satisfies equation 27, it is the leastsquares migration image. In equations 26 and 27, W is a dataweighting matrix allowing for incomplete data, and ␮ is a trade-off parameter. We solve equation 27 by the conjugate gradient method, and implicit construction of the matrices A, AH and W. This is a common idea used in inversion and is discussed for example, in Claerbout 共1992兲 where linear operators are built for the conjugate gradient method without the explicit construction of matrices. The cost function in equation 26 consists of two parts. The first term on its right-hand-side is the data-misfit function. It describes how well the data, modeled by the forward operator and some instance of m, fit the observed data dobs. The second term on the righthand-side of equation 26 is the model-norm, and is an example of Tikhonov regularization 共e.g., Tikhonov et al., 1995兲. The role of the

EXAMPLE: FOUR-LAYER ACOUSTIC MODEL For our first example, we consider the synthetic data in Figure 2b. The data are generated using an acoustic finite difference code with absorbing boundary conditions 共including no free-surface兲, and the acoustic four layer model in Figure 2a. We collect shots spaced,

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Figure 2. Four-layer acoustic model example showing, 共a兲 the velocity model, 共b兲 the data, 共c兲 the adjoint 共migration兲, and 共d兲 the inverse 共leastsquares migration兲. The adjoint in 共c兲 is shown after summing over all shots, and both 共c兲 and 共d兲 are plotted using a 50% clip.

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laterally, every 20 m, and at a depth of 150 m. The geophones are spaced every 5 m and are also at a depth of 150 m. In total, there are 71 shots and 281 geophones per shot. We add Gaussian random noise to the data so that the signal-to-noise ratio is 3, and, at random, we replace 60% of the traces with dead 共zero兲 traces. Due to the missing traces, we let W in equation 27 be a diagonal matrix such that its ith diagonal element is,



1, i 苸 Id 0, i 苸 Id,



EXAMPLE: SIGSBEE 2A DATA

共28兲

For our second example, we consider the Sigsbee 2a model from the SMAART JV project 共Bergsma, 2007兲. We use finite-difference data obtained from the Madagascar project 共Irons, 2007兲. The portion of the model that we consider is shown in Figure 4a, and Figure 4b is the velocity model used for migration. Figure 6a plots the oneand-only shot gather that we use in this example. The shot is located at 3.33 km, and there are 348 receivers. The near-offset receiver is coincident with the shot, and the far-offset receiver is located at 11.26 km 共offset 7.93 km from the shot兲. The shots and receivers are placed at a depth of 7.62 m. The aperture for the migration runs from the shot location out to 14.3 km 共offset 10.97 km from the shot兲. The results of applying the adjoint and inverse to the shot gather in Figure 4c are shown in Figure 5. In Figure 5a and b we show, respectively, the adjoint and the inverse, and in Figure 5c we show the true reflectivity model, which is representative of the scattering potential ␣ 共for display purposes, we applied a low-pass filter to the reflectivity兲.

where, in equation 28, Id is the set of indices corresponding to dead traces in d. We apply the adjoint 共split-step shot-profile migration兲 to each common shot gather, giving common shot image gathers, and then sum over the shot dimension, giving the result in Figure 2c. We apply the inverse 共shot-profile least-squares migration兲 to give Figure 2d. The differences between Figures 2c and d are 共in travel time兲 small, but there are differences in amplitude that make the reflectors more visible in the inverse, as compared to the adjoint. We apply the forward 共de-migration兲 operator, for one shot location 共Figure 3a兲, to the migrated image found by inversion 共Figure 3b兲. Figure 3a is the corresponding shot gather in the input data, and Figure 3c is its noise-free counterpart. Finally, Figure 3d is the difference between the reconstructed data and the noise-free data. The

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45.72 m, with the last shot located at 14.3 km. Each shot has 348 receivers, and the survey follows a towed-streamer geometry. Figure 7a is the sum over the 241 common shot-image gathers that are computed using migration. Figure 7b is the corresponding least-squares migration result. We note that Figures 7a and b are plotted using the same percentile clip. Least-squares migration shows some uplift in amplitude and resolution compared to migration, the differences concerning reflector locations and point diffractor resolution are,

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In the inversion, we note that the split-step wavefield propagator makes heavy use of the Fourier transform that tends to cause edge effects. We allow for these by including a taper in the data-weights W. This means that we do not require the least-squares migration algorithm to fit far offset traces. The effect of W is visible when reconstructing the data gather from the least-squares common shot image gather, and is illustrated in Figure 6. Figure 6a plots the shot gather, and Figure 6b plots the shot gather constructed using the migrated gather in Figure 5b, and the wavefield modeling operator given by equations 16 and 17. Due to the limited data 共single shot gather兲, the migrated images are, in places, of poor quality, especially where the offset from the source is large. We observe that the inverse compared to the adjoint has, in places, improved the image. In particular, artifacts have been reduced, and the point diffractors, especially those at 5-km depth and far offset, are better resolved. We include the single shot example because it illustrates the difference between migration and leastsquares migration under limited data. In this case the limitation is, admittedly, extreme 共a single shot gather兲, and is not adequate for illumination of the entire subsurface. Of course, when more shots are used to generate the image, the resolution improves, and we show this in Figure 7 where 241 shot gathers are used. The first shot is the same as the single-shot experiment, located at 3.33 km, and with the same geophone configuration. The remaining shots are spaced every

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Figure 4. Sigsbee 2a example: 共a兲 velocity model, and 共b兲 migration velocity model.

Figure 5. Sigsbee 2a example, single shot: 共a兲 the migration 共adjoint兲, 共b兲 the least-squares migration 共inverse兲, and 共c兲 the true reflectivity.

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Kaplan et al. gradient algorithm allows us to solve the least-squares normal equations with the forward/adjoint operators, and without the explicit storage of the matrices or their transpose matrices. We note that some authors 共e.g., Valenciano et al., 2009兲 take an alternative and interesting approach, coping with the size of the Hessian matrix AHA by observing that its main contributions occur close to its diagonal, therefore, allowing for its sparse representation. The methods in this paper are developed using the split-step wavefield propagator. It is, therefore, subject to error for large angle plane-wave contributions to the wavefield when there is strong lateral variation in the migration velocity model. This is exhibited in the Sigsbee 2a example, and in particular, in the imaging of the reflectors that fall within the vicinity of the salt body. We point out that using more accurate propagators such as those used in extended split-step 共Kessinger, 1992兲 or phase-shift plus interpolation 共PSPI兲 migration 共Gazdag and Sguazzero, 1984兲 will, undoubtedly, lead to more accurate algorithms and better-resolved migration results. In the case ofPSPI, the corresponding multiple reference slowness 共MRS兲 logicwould change the approximation of the Green’s function and all subsequent steps of the derivation, accordingly. The benefit of using more comprehensive estimates of the Green’s function is, of course, more accurate migration results, but at the expense of algorithmic complexity. We illustrated migration results in depth and lateral space coordinates. We recognize that it would be equally beneficial to show results in amplitude versus ray-parameter 共AVP兲 gathers. Previous authors have made note of the advantage of least-squares migration in correcting amplitudes in AVP gathers. For example, Kühl and Sacchi 共2003兲 use AVP gathers for source-receiver least-squares migration. Moreover, we note that Rickett and Sava 共2001兲 present an algorithm for computing AVP gathers that is applicable to shot-profile migration. To accommodate the construction of AVP gathers, one would use the previously mentioned alternative parameterization of model space in which the forward operator A excludes the sum over shots.

perhaps, less tangible than the single shot example, but are still evident, especially in the sub-salt region.

DISCUSSION In building the least-squares system of equations, we have made some choices. We choose to represent the scattering potential ␣ with a sum over prestack migrated shot gathers. One could alter the forward operator A to exclude the sum over shots. This would increase the degrees of freedom in the model space, allowing a better fit to the data; however, we found that the joint model space produces a higher quality image. In addition, the choice produces a more efficient inversion because the smaller model space allows the conjugate gradient algorithms to converge in fewer iterations. We are tasked with making these choices exactly because of the shot-profile parameterization of the scattering potential. If, on the other hand, we choose a source-receiver parameterization of the scattering potential, then data are required on some rectilinear grid in shot and receiver 共or midpoint and offset兲 coordinates. This is to accommodate the 4D Fourier transforms required by the source-receiver split-step 共de-兲 migration algorithm. Moreover, the model space becomes less physical, as the offset coordinate does not have a one-to-one correspondence with the earth model. Instead, it must be interpreted in terms of the source-receiver imaging condition that states that energy migrates to where source and receiver are coincident or offset is null. The matrices in the least-squares normal equations 27 are large. For the Sigsbee 2a example with 241 shots, A has approximately 200 million rows and 1 million columns. This makes its explicit computation and storage infeasible. Instead, the construction is implicit so that, for example, given some m, we compute d ⳱ Am using an operator that implements equations 8 and 14. Likewise, given some d,we compute m† ⳱ AHd using an operator that implements equations 20 and 21. We verify that our operators make a forward/adjoint pair using the dot-product test 共e.g., Claerbout, 1992兲. The conjugate

Figure 6. Sigsbee 2a example, single shot: 共a兲 the shot gather, 共b兲 the reconstructed shot gather, 共c兲 the difference between 共a兲 and 共b兲. For plotting, the amplitudes are clipped to 10 percent of the their maximum.

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-squares migration algorithm. This differs from previously published source-receiver least-squares migration algorithms, allowing for the 共de-兲 migration operators to be applied one shot at a time, as well as giving an alternative parameterization of the migrated image gathers. We applied migration and least-squares migration to data generated from the Sigsbee 2a model, showing differences between the two results. For example, the point scatterers in the Sigsbee 2a were better resolved, especially when the input data consist of a single shot gather. In general, least-squares migration has produced more balanced amplitudes and a slight improvement to resolution. Further, we note that least-squares migration is capable of working under noisy input data by making use of prior constraints 共model norm and data weights兲 in the inversion.

ACKNOWLEDGMENTS 13

2

6

We acknowledge the SAIG consortium members for continued support, ConocoPhillips for a summer internship given to Sam Kaplan, and the WestGrid computing facilities used in generating the examples. We thank the SMAART JV for the Sigsbee 2a model. We thank the reviewers and, especially, the associate editor, John Etgen, for constructive suggestions and comments that improved the quality of the manuscript and its examples.

8

APPENDIX A

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DERIVATIONS FOR SHOT-PROFILE WAVEFIELD MODELING

c)

Lateral position (km) 4

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We derive ␺ s共1兲 in equation 9 and ␺ s共2兲 in equation 13. First, we consider ␺ s共1兲. We substitute the Green’s function in equation 4 into the Born approximation in equation 2, taking the support of ␣ below the measurement surface and within D1,

Depth (km)

4

␺ s共1兲共xg,z0兩xs,z0;␻ 兲 ⳱

冉 冊冉 冊 1 2␲

6



4

2

c1共1兲

z1 ⬁

⫻ 8

冕冕

ⳮ1

ⳮ1

ei␻ 共c0共1兲共xg兲ⳮc1共1兲兲共z⬘ⳮz0兲

z0 ⳮ⬁

冕冉 ⬁

Figure 7. Sigsbee 2a example, 241 shots: 共a兲 the adjoint 共migration兲, 共b兲 the inverse 共least-squares migration兲, and 共c兲 the true reflectivity.

CONCLUSIONS We show, in detail, the derivation of shot-profile wave-equation migration and de-migration operators from the Born approximation, and split-step wavefield modeling. We derived the forward 共de-migration兲 operator and its adjoint 共migration兲. From this analysis, we recognize the usual shot-profile migration algorithm involving the downward continuation of the source and receiver side wavefields, along with an imaging condition. Given the forward and adjoint operators for wave-equation shotprofile 共de-兲 migration, we built a shot-profile wave-equation least-





ⳮ⬁

1 i4kgz共1兲



⫻eⳮikgx·共x⬘ⳮxg兲eikgz共1兲共z⬘ⳮz0兲dkgx ⳮ1

ⳮ1

⫻␣ 共x⬘,z⬘兲ei␻ 共c0共1兲共x⬘兲ⳮc1共1兲兲共z⬘ⳮz0兲

冕冉 ⬁



ⳮ⬁



1

⬘ i4kz共1兲



⬘ 共z⬘ⳮz0兲 eikx⬘·x⬘eikz共1兲

⫻g共kx⬘,␻ ;xs兲dkx⬘dx⬘dz⬘ .

共A-1兲

In equation A-1, we have recognized an expression for the synthetic source used in shot-profile migration algorithms,

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g共kx⬘,␻ ;xs兲 ⳱ f共␻ 兲eⳮikx⬘·xs .

␺ 共2,1兲共x,z兩xs,z0;␻ 兲 ⬁

We rearrange terms and regroup integrals in equation A-1, giving the equivalent expression,

⳱ f共␻ 兲



G0共1兲共x⬘,z1兩xs,z0;␻ 兲G0共2兲共x,z兩x⬘,z1;␻ 兲dx⬘

ⳮ⬁ ⬁ z2

␺ s共1兲共xg,z0兩xs,z0;␻ 兲 ⳱

冉 冊冉 冊冕 ␻

2

c1共1兲 ⬁





e

4

1 2␲

ikgx·xg

ⳮ⬁



z1

ⳮ1

ⳮ1

ei␻ 共c0共1兲共xg兲ⳮc1共1兲兲共z⬘ⳮz0兲





ⳮ1

共A-5兲

We recognize that the second term on the right-hand-side of equation A-5 contains the scattered portion of the wavefield which, here, we denote as ␺ s共2,1兲. Recognizing the recursion in equation A-5 allows us to write ␺ s共2,1兲 to first order in ␣ ,

1 ⳮ eikgz共1兲共z⬘ⳮz0兲 i4kgz共1兲

ⳮ1

␺ s共2,1兲共x,z兩xs,z0;␻ 兲

eⳮikgx·x⬘ei␻ 共c0共1兲共x⬘兲ⳮc1共1兲兲共z⬘ⳮz0兲



ⳮ⬁

冕 冉 ⬁



␻2 2 c0共2兲

⫻ ␣ 共x⬘,z⬘兲␺ 共2,1兲共x⬘,z⬘兩xs,z0;␻ 兲dz⬘dx⬘ .





G0共2兲共x,z兩x⬘,z⬘;␻ 兲

ⳮ⬁ z1

z0



冕冕

e

ikx⬘·x⬘

ⳮ⬁



1

⬘ i4kz共1兲



⬇ f共␻ 兲 e

ikz共1兲 ⬘ 共z⬘ⳮz0兲

⫻ g共kx⬘,␻ ;xs兲dkx⬘␣ 共x⬘,z⬘兲dx⬘dkgxdz⬘ .

冕冕

G0共1兲共x⬙,z1兩xs,z0;␻ 兲

ⳮ⬁

z2



共A-2兲

Finally, we make a change of variables in equation A-2, from kx⬘ to kgx, and x⬘ to xg, and recognize three 2D Fourier transforms to give equation 9. Note that the change of integration variables to lateral geophone 共space and wave-number兲 is somewhat arbitrary. In practice, we extend xg beyond the aperture of the geophones by adding zero-traces to either side of the shot gather. This, in effect, makes the aperture of ␣ for shot xs 共the migrated image shot gather兲 independent of the recording aperture. Next, we consider ␺ s共2兲, the contribution to the scattered wavefield from the second layer denoted by D2. In equation 9, ␺ s共1兲 is a first order solution to the perturbed Helmholtz equation,



G0共2兲共x,z兩x⬘,z⬘;␻ 兲

z1

␻2 2 ␣ 共x⬘,z⬘兲 c0共2兲

⫻ G0共2兲共x⬘,z⬘兩x⬙,z1;␻ 兲dz⬘dx⬙dx⬘ .

共A-6兲

Equation A-6 is the scattered wavefield in D2 due to the scattering potential ␣ 苸 D2. To find the scattered wavefield at the measurement surface due to ␣ 苸 D2, we use ␺ s共2,1兲 as a boundary condition at the bottom surface of D1 共neglecting reflected energy兲. Then, we let ␺ s共2兲 be the scattered wavefield in D1 due to ␣ 苸 D2. In particular, we have,

L共␺ s共2兲,c0共1兲兲 ⳱ ␺ s共2,1兲共x,z兩xs,z0;␻ 兲␦ 共z ⳮ z1兲, 共A-7兲 so that for z 苸 D1,

L共␺ s共1兲,c0共1兲兲 ⳱ f共␻ 兲␦ 共x ⳮ xs兲␦ 共z ⳮ z0兲

␻2 Ⳮ 2 ␣ 共x,z兲␺ 共x,z兩xs,z0;␻ 兲, 共A-3兲 c0共1兲 where L in equation A-3 is the Helmholtz operator. The background wavefield in equation 1 propagates the wavefield 共without scattering兲 from the source through to the bottom of D1, and can be used as a boundary condition at the top of D2. Translating this boundary condition into a forcing term gives for D2,

␺ s共2兲共x,z兩xs,z0;␻ 兲 ⬁

⳱ f共␻ 兲

冕冕 冕

G0共1兲共x,z兩x⬘,z1;␻ 兲

ⳮ⬁

z2



冕 z1

G0共2兲共x⬘,z1兩x⬙,z⬙;␻ 兲

␻2 2 ␣ 共x⬙,z⬙兲 c0共2兲

⫻ G0共2兲共x⬙,z⬙兩x⵮,z1;␻ 兲dz⬙ L共␺ 共2,1兲,c0共2兲兲 ⳱ f共␻ 兲G0共1兲共x,z兩xs,z0;␻ 兲␦ 共z ⳮ z1兲

␻ 2 ␣ 共x,z兲␺ 共2,1兲共x,z兩xs,z0;␻ 兲, c0共2兲

⫻ G0共1兲共x⵮,z1兩xs,z0;␻ 兲dx⵮dx⬙dx⬘,

共A-8兲

2



so that,

共A-4兲

which can be made equivalent to equation 14 for ᐉ ⳱ 2. To show this, we evaluate it at the measurement surface 共xg,z0兲, and substitute for the Green’s function in equation 4 so that after re-grouping integrals, we find,

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L.S. shot-profile split-step migration

␺ s共2兲共xg,zg兩xs,zs;␻ 兲

冉 冊 冕

1 ⳱ 2␲

8

␻2 2 c1共2兲

z2

us共1兲共xg,z1;␻ 兲

z1

冕 冕 冕



eikgx·xg

冕 冕 冕

eikx⬘·x⬘

⫻u p共1兲共kgx,z1;␻ 兲

eⳮikgx·x⬘us共2兲共x⬘,z⬙;␻ 兲

⫻u p共2兲共kx⬘,z⬙;␻ 兲

eⳮikx⬘·x⬙ us共2兲共xg,z⬙;␻ 兲

eikx⬙·x⬙

eⳮikx⬙·x⵮us共1兲共x⵮,z1;␻ 兲

eikx⵮·x⵮

⫻u p共2兲共kx⬙,z⬙;␻ 兲



⫻u p共1兲共kx⵮,z1;␻ 兲g共kx⵮,␻ ;xs兲dkx⵮dx⵮dkx⬙ ⫻␣ 共x⬙,z⬙兲dx⬙dkx⬘dx⬘dkgxdz⬙,



共A-9兲

where we have, again, used the simplifying assumption in equation 12. Additionally, to avoid notational clutter, we have used the definitions in equations 10 and 11, and where the limits of integration are not specified, they are 共ⳮ⬁,⬁兲. In equation A-9, we make a change of integration variables from each of x⬘, x⬙ and x⵮ to xg, and from each of kx⬘, kx⬙ and kx⵮ to kgx. Then, recognizing four 2D inverse Fourier transforms, and three 2D Fourier transforms gives equation 14 for ᐉ ⳱ 2.

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Derivation of forward and adjoint operators for least ...

tic model, and the second uses data from the Sigsbee 2a model. INTRODUCTION. We derive and implement operators for shot-profile migration and.

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