GEOPHYSICS, VOL. 68, NO. 4 (JULY-AUGUST 2003); P. 1348–1356, 9 FIGS. 10.1190/1.1598128

3D ray+Born migration/inversion—Part 1: Theory

Gilles Lambare´ ∗ , Stephane ´ Operto‡ , Pascal Podvin∗ , and Philippe Thierry∗ to cure potential amplitude artifacts and improve spatial resolution. This last point is the main focus of this pair of paper and its companion paper (Operto et al., 2003). The theory of ray-based quantitative migration was developed more than a decade ago (Beylkin, 1985; Bleistein, 1987; Beylkin and Burridge, 1990; Jin et al., 1992). It relies either on the ray+Born or on the ray+Kirchhoff linearized approximations. These approximations differ by their description of the “reflecting/diffracting” components of the model (i.e. in terms of impedance and density perturbations or in terms of specular reflectivity, respectively). The associated formulas and numerical implementations are very close, and it is not so easy to discriminate between them in the context of seismic imaging (Beydoun and Jin, 1994). Some significant differences remain, however, in their practical algorithms. Ray+Born migration/ inversion provides impedance and density perturbations in a single step, whereas ray+Kirchhoff migration/inversion requires some postprocessing of the common image gathers (CIGs) in order to provide an equivalent result. A two-step strategy has many advantages. In particular, it opens ways to mitigate the effects of incomplete illumination or incorrect migration velocities (via migration velocity analysis). Whatever the approach, the proper choice of the domain used to compute CIGs has well-known impact on the quality of the results. In the Kirchhoff approach, this amounts to choosing properly minimal subdata sets to improve CIG reliability. CIGs in the offset domain are the most frequently used, but the computation of CIGs in the diffracting/reflecting angle domain seems to exhibit advantages in complex media (Xu et al., 2001). In these papers, we focus on the single-step ray+Born approach, where similar questions arise, but are reformulated in terms of improving the choice of the stacking domain and the design of an optimal weighting of the migration stack with both resolution and amplitude reliability in mind. The question is intricate, however; for instance, resolution could be improved by favoring the short offset contributions (less affected by stretching and velocity errors), but this would compromise the signalto-noise ratio for AVO estimation, where the contribution of large offset is essential.

ABSTRACT

Prestack ray+Born migration/inversion can be split in two steps : the computation of common image gathers (CIGs) and their weighted stack (the migration stack). The choice of the domain for the CIGs (shot, offset, angle, etc.) has a direct impact on the resolution of the migration stack. This resolution can be studied easily in the frame of ray+Born migration/inversion theory resulting into improved migration/inversion formulas according to the acquisition geometry. This paper is devoted to this analysis in the cases of a simple 2D acquisition and of a 3D swath acquisition, both corresponding to classical data sets from the SEG/EAGE 3D overthrust experiment. We show that the migration formula originally designed for 3D marine acquisition is not adaptable to the 3D swath acquisition. Finally, we propose a new formula for this specific acquisition, which improves the resolution of the final migrated image. The relevance of this new formula is illustrated in the frame of the SEG/EAGE experiment in the companion paper.

INTRODUCTION

In the context of 3D seismic imaging, ray-based migrations are particularly appreciated for their CPU efficiency and for their ability to provide a quantitative estimation of the physical properties of the reflectors. Three-dimensional applications of quantitative ray-based migration (migration/inversion) have already been presented as feasability studies (Sevink, 1996; Clochard et al., 1997a,b; Tura et al., 1998; Thierry et al., 1999b). The capacity of the approach, in particular for improving amplitude variation with offset/amplitude variation with angle (AVO/AVA) attribute estimation on real cases, is currently under examination (Baina et al., 2001; Lenain et al., 2001). Its success depends on accurate control of the preprocessing sequence and refinements of the migration/inversion strategy

Manuscript received by the Editor March 11, 2002; revised manuscript received December 12, 2002. ∗ ´ Ecole des Mines de Paris, Centre de Recherche en Geophysique, ´ 35 rue Saint Honore, ´ 77 305 Fontainebleau Cedex, ´ France. E-mail: lambare@ geophy.ensmp.fr; [email protected]; [email protected]. ´ ‡Formerly Ecole des Mines de Paris, Centre de Recherche en Geophysique, ´ 35 rue Saint Honore, ´ 77 305 Fontainebleau Cedex, ´ France; presently UMR Geosciences ´ Azur, CNRS-UNSA, 06235 Villefranche-sur-mer, France. E-mail: [email protected]. ° c 2003 Society of Exploration Geophysicists. All rights reserved. 1348

3D Ray+Born Inversion—Part 1

In this paper, we show, through discussion of the migration/ inversion resolution filter, that the theory of ray+Born migration/inversion provides a correct frame to help make sound choices according to the acquisition geometry. This is illustrated on two acquisition geometries : a standard 2D multichannel line and a 3D swath acquisition, for which optimal migration/inversion formulas are established. In the companion paper (Operto et al., 2003), these formulas are applied to two subdata sets of the 3D SEG/EAGE overthrust experiment (Aminzadeh et al., 1997). 3D RAY+BORN MIGRATION/INVERSION

We consider a single-parameter acoustic medium parameterized by velocity c(x). Using the Born approximation, one can obtain a linear relation between the diffracted wavefield δG(r, ω; s) (s and r denote the source and the receiver positions and ω the angular frequency) and the squared slowness perturbations δ`(x) = −(1/c03 (x))δc(x) (the model parameters) (e.g, Thierry et al., 1999a,b):

ZZZ δG(r, ω; s) = B[δ`] =

M

dxB(r, x, ω; s)δ`(x). (1)

between the observed and the computed wavefields by a quasiNewton algorithm. Consider a 3D multichannel data set parameterized by parameter set LSR (line, shot, and receiver indices for a 3D marine acquisition). The classic solution of the inverse problem is given in compact form by [see, for example, Tarantola(1986)]

δ` = [B† QB]−1 B † QδG,

B(r, x, ω; s) = S(ω)A(r, x, s)eiωT (r,x,s) ,

(2) where

where S is a source term, A is the amplitude term of the Born operator (product of the amplitudes associated with rays r → x and s → x), and T is the two-way traveltime of the scattered ray path s → x → r (Figure 1). We have in three dimensions (Thierry et al., 1999b),

  

S(ω) = ω2 , A(r, x, s) = A(r, x)A(s, x),   T (r, x, s) = T (r, x) + T (s, x).

(3)

R

T(s,x),A(s,x)

T(r,x),A(r,x) x

θ = βs− β r ψ = ( βs + β r )/2

LSR

(

ω

dωQD(r, x0 , s, ω)e−ik·(x−x0 ) , (5)

D(r, x0 , s, ω) = |S(ω)|2 |A(r, x0 , s)|2 , k(r, x0 , s, ω) = ω ∇x0 T (r, x0 , s),

where | denotes the norm (or absolute value); k is the wavenumber vector defined also by (Figure 1)

k = ωq(r, x0 , s) = ω[p(x0 , r) + p(x0 , s)].

δ(x − x0 ) =

1 (2π )3

ZZZ

+∞ −∞

dke−ik·(x−x0 ) ,

(7)

(8)

and the asymptotic approximation of the Hessian [equation (5)]. In order to formulate the Hessian as a weighted Dirac function, one has to apply a change of variables, so that the components of k are introduced as some of the variables of integration of the Hessian:

(LSR, ω) −→ (k, ξ ).

βr

(6)

(9)

p

θ

p

r

Z dLSR

Jin et al. (1992) recognized the analogy between the Fourier integral expression of the Dirac function,

Following Jin et al. (1992), the model perturbations δ`(x) are obtained by least-square minimization of the weighted misfit

S

(4)

where Q is the weight of the quasi-Newtonian inversion and the superscript† denotes the adjoint operator. The operator B† QB is the Hessian, and (−B † QδG) is the gradient of the misfit function. Since the Hessian is an operator mapping the model space to itself, it represents a huge square matrix whose dimension equals the square of the number of scatterers in the migrated image. In order to compute its inverse, Jin et al. (1992) proposed that Q be chosen in such a way that a diagonal asymptotic approximation of the Hessian exists. The Hessian involves an oscillatory integral whose stationary points are on the diagonal. One can keep the leading asymptotic terms of the amplitude and phase of the Hessian around the diagonal, x = x0 [equation (29) in Jin et al. (1992)]:

H(x, x0 ) ZZZ =

B is the asymptotic linear Born operator given by

1349

s

q

βs

Ψ

FIG. 1. Ray parameters involved in the 2D ray+Born migration/inversion. Note that the angles are oriented.

For a usual 3D marine multichannel acquisition, the data space (LSR, ω) is a 4D space. The wavenumber k parameterizes the 3D model space. In order to have a one-to-one mapping, one must introduce an extra parameter ξ which accounts for data redundancy (namely, the extra dimension of the data space compared to the model space). The choice of ξ is particularly important because it controls how this data redundancy is taken into account by the ray-Born migration/inversion, and it can be

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Lambare´ et al.

used as the parameter for migration velocity analysis. The parameter ξ must be independent of k = ωq. Until now, it was always chosen independent of ω and connected with the aperture of rays at the diffracting point. For example, a similar approach can be developed in two dimensions, in which ξ was chosen to be the diffraction angle θ (Jin et al., 1992; Forgues, 1996; Thierry et al., 1999a) (Figure 1). In another 3D context (Thierry et al., 1999b), ξ was chosen to be |q| (Figure 1). We shall discuss later the influence of the choice of ξ and see how this choice can be improved, driven by the acquisition geometrical pattern. Note that the extra dimension in the data space results from the fact that the full multichannel dataset is processed in one step. This is a major difference from Kirchhoff-type prestack depth migration, in which the data are sorted by commonparameter gathers (generally, common shot or common offset). After applying the change of variables (9) to equation (5), we have

Z

H(x, x0 ) =

ZZZ

dkQD(r, x0 , s, ω)

dξ

× J (r, x0 , s, ω)e−ik·(x−x0 ) , where

¯ ¯ ¯ ∂(LSR, ω) ¯ ¯ ¯ J (r, x0 , s, ω) = ¯ ∂(k, ξ ) ¯

(10)

H(x, x0 ) =

1 (2π)3

ZZZ

dξ

dke−ik·(x−x0 ) ,

(12)

(13)

and can be approximated by a weighted Dirac function (Jin et al., 1992)

H(x, x0 ) ≈

[ξ ]max min (x0 )δ(x

− x0 ),

with

(14)

where [ξ ]max min (x0 ) = ξmax (x0 ) − ξmin (x0 ) denotes the difference between the minimal and maximal values of ξ at x = x0 provided by the acquisition geometry. After the diagonalization of the Hessian and introduction of the expression for Q [equation (12)] into the gradient, the final 3D ray+Born migration/inversion formula reduces to a weighted diffraction stack. The derivation of the formula depends on the choice of ξ , which allows various simplifications of the Jacobian [equation (11)]. For example in three dimension

X 1 E(r, x, s)δG(r, s, T (r, x, s)) max [|q|]min LSR

¯ ¯ 1 |q|1LSR ¯¯ ∂(q) ¯¯ E(r, x0 , s) = (2π )2 A(r, x0 , s) ¯ ∂(LSR) ¯

(15)

(16)

(1LSR being the increment in the trace parameterization), and where Thierry et al. (1999b) used

¯ ¯ ¯ ¯ ¯ ¯ ¯ ∂(|q|, k) ¯ ¯ = |ω|2 |q|¯ ∂(q) ¯. ¯ ¯ ∂(LSR) ¯ ¯ ∂(LSR, ω) ¯

(17)

Derivation of the 2D or 2.5D migration/inversion formula relies on the same principles as the 3D analysis (Lambare´ et al., 1992; Jin et al., 1992; Forgues, 1996; Thierry et al., 1999a). In 2.5 dimensions, the diffraction angle θ and |q| = 2 cos(θ/2)/c(x) are two natural candidates for ξ (Figure 1). Both choices provide a diffraction stack formula differing by the stacking weight Eξ :

(11)

After inserting this expression for Q, the Hessian becomes

Z

δ`(x) =

δ`(x) =

denotes the Jacobian of the relation connecting the data space (LSR, ω) to the migrated image space (k, ξ ) (i.e., the absolute value of the determinant of the Jacobian matrix). In equation (10), the bounds of integration over the parameters k and ξ are controled by the bandwidth of the source and the acquisition geometry. They will control the resolution of migration/ inversion, as we will see later. Jin et al. (1992) proposed that the weight Q should be chosen to make the amplitude term QDJ in the integral [expression (10)] reduce to 1/(2π)3 in order to match the integral expression (8) of the Dirac. We obtain

1 Q= . 3 (2π) J (r, x0 , s, ω)D(r, x0 , s, ω)

and for ξ = |q|, Thierry et al. (1999a) obtained

1 X Eξ (r, x, s)(F ∗ δG)(r, s, T (r, x, s)) [ξ ]max min SR (18)

with

H (−t) , F(t) = √ −π t

(19)

where H (t) denotes the Heaviside function. We have for ξ = |q|,

E|q| (r, x, s) =

¯ ¯ ¯ ∂(q) ¯ 1 1s1r ¯ ¯, 2π A2.5D (r, x, s) ¯ ∂(sr) ¯

(20)

and for ξ = θ,

¯ ¯¯ ¯ |q|1s1r ¯¯ ∂(βs ) ¯¯¯¯ ∂(βr ) ¯¯ 1 , Eθ (r, x, s) = 2π A2.5D (r, x, s) ¯ ∂(s) ¯¯ ∂(r) ¯

(21)

where 1s and 1r denote the shot and receiver position steps, respectively, βs and βr the angles at x of the rays r → x and s → x (Figure 1), and A2.5D the amplitude of the 2.5D forward Born operator (Thierry et al., 1999a). RESOLUTION ANALYSIS

When approximating the Hessian as a weighted Dirac function [equation (14)], we assumed that k was covering the whole 3D space [namely, that the bounds of integration over k in equation (13) were infinite]. It is obviously not the case for real acquisition conditions, where the bandwidth of the source and the acquisition geometry are limited. For simplicity, we shall now analyze the influence of a band-limited source and limited acquisition geometry in the 2D case. The 2-D Fourier integral expression of the Dirac can be written in polar coordinates (|k|, ψ) [ψ is the dip angle (Figure 1)]:

δ(x − x0 ) =

1 (2π )2

Z

Z

d|k|

dψ|k|e−ik·(x−x0 ) .

(22)

3D Ray+Born Inversion—Part 1

Then, the 2D asymptotic Hessian can be written without approximation:

Z H(x, x0 ) =

Z

ξmax

ψmax (ξ )

dξ

dψ

ξmin

Z

×

ψmin (ξ ) |k|max (ξ,ψ)

|k|min (ξ,ψ)

d|kkk|e−ik·(x−x0 ) .

(23)

We define a truncation filter Tx0 (ξ, x) for each value of ξ :

Z

H(x, x0 ) = δ(x − x0 ) ∗ where

Z

Tx0 (ξ, x) =

Z

ψmax (ξ )

ξmax ξmin

dξ Tx0 (ξ, x),

|k|max (ξ,ψ)

dψ ψmin (ξ )

|k|min (ξ,ψ)

(24)

d|k||k|e−ik·x , (25)

and the average filter Rx0 (x):

H(x, x0 ) = [ξ (x0 )]max min δ(x − x0 ) ∗ Rx0 (x), where

Rx0 (x) =

1 [ξ ]max min

Z

ξmax ξmin

dξ Tx0 (ξ, x)

(26)

(27)

is an average over ξ of the individual Tx0 (ξ, x). Thus, the Hessian operator H is not a Dirac function, as suggested by the approximated expression of equation (14), but a filtered Dirac function, where Rx0 (x) is the filter, described by equation (27). We must remember that there is a component of redundancy in prestack data when used for migration. In fact, our approach of ray+Born migration/inversion performs an average of partly redundant constant-ξ migrated images. Each constant-ξ migrated image is self consistent in the sense that it provides directly a quantitative estimation of the model perturbations (in a similar way as individual common-offset migrated images). The choice of ξ controls how the decomposition in constant-ξ migrated images is done and the way they are averaged. It also provides the domain in which the CIGs are displayed. Consider an individual constant-ξ migrated image associated with a given truncation filter Tx0 (ξ, k). The double truncation (in dip and amplitude) of the Dirac function limits the spatial resolution of the image because the k-space is only partly covered. The partial coverage of |k|, resulting mainly from the limited bandwidth of source |k| = ω|q|, limits the possibility of separating closely-spaced model perturbations in the migrated image. Note that when aperture increases, |q| decreases and the spatial resolution shifts to lower spatial frequencies. The partial coverage of the ψ domain, resulting from the characteristics of the acquisition geometry, limits the dips that we can image (Miller et al., 1987). The global truncation filter Rx0 (k) [equation (27)] is not a pure truncation filter, but a smooth filter. In the final ray+Born migration/inversion formula, we assume that Rx0 (x) ≈ δ(x) (i.e., Rx0 (k) ≈ 1), and consequently the stacked final migrated image [equation (15)] corresponds to the exact perturbation model filtered by Rx0 (x), which is the local spatial resolution filter. At a given x0 , the zone of perfect resolution in the k-space is limited to the one resulting from the common overlapping of all the individual nonzero Tx0 (ξ, k). The other k-components

1351

of the migrated image are all underestimated when using our approximation of the Hessian [equation (14)]. Several iterations of ray+Born migration/inversion were initially proposed for improving the resolution in the spectral domain, where the spatial spectral component of the approximated asymptotic Hessian are not exactly zero (Forgues, 1996; Sevink, 1996; Jin et al., 1992; Lambare´ et al., 1992). In three dimensions, for realistically-sized applications, iterations are computationally expensive, and one attempt should be made to improve the migration/inversion formula to enhance resolution at once. From all these considerations, we can derive some rules to guide our choice of the ξ parameter. A good choice for ξ will be one providing a good resolution filter Rx0 (x) over the entire image while retaining the benefits of the stack for noise reduction. First, ξ must provide individual constant-ξ migrated images with maximum resolution. Second, it should also provide the most uniform Tx0 (ξ, k) series over ξ in such a way that the normalized stack Rx0 (k) does not alter too much the final image resolution. Third, it must favor, among all the individual constant-ξ migrated images, those with the maximum resolution. Moreover, remembering that the Born approximation is valid for small diffraction angles θ (Figure 1), the ξ parameter must also be chosen such that the individual constant-ξ migrated images associated with small aperture are favored in the summation. ILLUSTRATION FOR 2-D AND 3-D ACQUISITION GEOMETRIES

2D acquisition A simple 2D canonical example will first illustrate these concepts. Consider a classical 2D marine acquisition geometry with a 4-km-long streamer and a near offset of 0 m and with shot positions ranging from [−∞, +∞]. Consider a point x0 located at a depth of 2 km. The velocity macromodel is homogeneous with a velocity of c = 3 km/s, and the bandwidth of the source signature is 5–22 Hz. The local truncation and resolution filters are compared for two expressions of ξ : the diffraction angle θ previously used for 2D applications by Jin et al. (1992), Forgues (1996), and Thierry et al. (1999a), and |q| (Figures 2 and 3). In both cases the spectral coverage varies from one ξ to the next (compare the constant-ξ truncation filters in Figures 2a–2d and 3a–3d), and the best coverage in terms of high spatial frequency and dip coverage is obtained at small aperture (small values of θ or high values of |q|). The resolution filter Rx0 (k) obtained with ξ = θ allows a better spectral coverage than the one obtained with ξ = |q|, suggesting that for 2D applications ξ = θ is the best choice (compare Figures 2e and 3e). Moreover, the choice of ξ = θ over ξ = |q| allows favoring small aperture diffracted raypaths, for which the Born approximation is more accurate. Indeed, the analysis of the local resolution filter Rx0 (k) was easily done in our 2D canonical case by using simple geometrical considerations and uniform infinite source coverage at the surface. We anticipate that the effects of the resolution filter will be more subtle for nonuniform 3D acquisition geometries. Among others, 3D applications will require the analysis of the k y crossline component in addition to k x and k z components. 3D swath acquisition We now apply our resolution analysis to a second data set, which corresponds to a swath acquisition of eight double lines,

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Lambare´ et al.

the source-receiver reciprocity for large offsets is not ensured at most of the midpoint positions along the line of sources (Figure 5). We consider the same source configuration (an infinite line of sources) and the same source bandwidth as in the 2D previous canonical example, but with a homogeneous velocity macromodel (c = 3 km/s). The k-space is now three dimensional, and we plot sections (Figue 6) of the local resolution filters Rx0 (k) for k y = 0 and k x = 0 and for a point just below the center of the receiver layout (point of type 5 in Figure 5). The shape of the k y = 0 section is similar to the one of the 2D case (compare Figures 3e and 6a). The small differences result from the

300-m apart without line overlap (Figure 4). A double line corresponds to two lines of 120 receivers, 50-m apart. This layout is applied to a single line of shots which corresponds to a swathtype acquisition. We presented an application of 3D ray+Born migration/inversion to this kind of data set in the frame of the 3D SEG/EAGE overthrust experiment in our companion paper (Operto et al., 2003). The swath acquisition has several properties which can cause footprints and amplitude artifacts on the migrated images. The first property is the nonuniform distribution of long offsets with source-receiver azimuth. Second, the long-offset coverage varies from one midpoint to the next (Figure 5). Moreover,

(a) -10

fx (km-1) 0

(b) 10

-10

0

fx (km-1) 0

10

-10 0

θ=0.deg fx (km-1) 0

(d) 10

θ=24.deg -10

10

0

0.5

-10

fz (km-1)

-10

fx (km-1) 0

scale

10

-10

fz (km-1)

-10

0

fz (km-1)

10

(c)

(e) 10

-10

fz (km-1)

fz (km-1)

-10

fx (km-1) 0

0

10

0

10

1.0

10

θ=48.deg

θ=72.deg

Resolution

FIG. 2. (a)–(d) Spectral resolution of four constant-θ truncation filters Tx0 (θ, k) for the 2D canonical acquisition geometry and model. In the gray scale, the black and white colors correspond to the optimal and the null resolutions, respectively. The labels of the axis f x and f z denote the spatial frequencies (k x = 2π f x ) andR (k z = 2π f z ). Note the variation of resolution over θ . (e) Spectral dθTx0 (θ, k). The labels for (a)–(d) are the same as for (e). resolution as described by the resolution filter Rx0 (x) = 1/[θ]max min

-10

fx (km-1) 0

10

(b)

0

fx (km-1) 0

10

0 -10

0

10

|q|=.667s/km -10

fx (km-1) 0

|q|=.591s/km

(d)

10

-10

fx (km-1) 0

10

0

0.5

-10

fz (km-1)

-10

fz (km-1)

10

-10

scale

10

(c)

(e)

-10

fz (km-1)

fz (km-1)

-10

-10

fx (km-1) 0

fz (km-1)

(a)

0

10

0

10

1.0

10

|q|=.533s/km

|q|=.648s/km

Resolution FIG. 3. (a)–(d) Spectral resolution of four constant-|q| truncation filters Tx0 (|q|, k) for the 2D canonical acquisition geometry and model. In the gray scale, the black and white colors correspond to the optimal and the null resolutions, respectively. The labels of the axis f x and f z denote the spatial frequencies (k x = 2π f x ) and (kR z = 2π f z ). Note the variation of resolution over |q|. (e) Spectral d|q|Tx0 (|q|, k). The labels for (a)–(d) are the same as for (e). resolution as described by the resolution filter Rx0 (x) = 1/[|q|]max min

3D Ray+Born Inversion—Part 1

nonuniform distribution of large offsets in the swath acquisition: (1) the zero dip (k x = 0) is not covered at large offsets; (2) some areas in the k-plane are covered only once due to the nonreciprocity of source-receiver pairs at large offsets. More importantly, the resolution in the k x = 0 sections is quite poor (because of the poor spectral resolution of the individual images associated with high values of |q|) (Figure 6b). The poor coverage of lateral dips when using ξ = |q| results mainly from the acquisition geometry (only one line of sources is available). For example, zero offset data corresponding to |q| = 2/c are only acquired along the shot line. The filter Rx0 (k) shows that, paradoxically, the amplitudes of laterally dipping events are understimated to a much greater extent when using this 3D formulation rather than a 2D one. We noticed also that the resolution of the migrated images may vary laterally. Indeed, we examine the sections of the resolution filter Rx0 (k) (Figure 7) for a point characterized by a strong asymmetrical distribution of long offsets (point of type 3 in Figure 5). Indeed, the asymmetrical coverage of large offsets modifies the resolution filter Rx0 (k) compared to the one of Figure 6 (obtained for a midpoint of type 5). We show in our companion paper (Operto et al., 2003) that these lateral variations of Rx0 (k) may be one cause of acquisition footprints in the migrated images. This analysis of the total truncation filter convinced us that the choice of ξ = |q| was not the optimal one for this data set. The ξ = |q| choice was poor not only in terms of resolution of inline dipping events (k x ), as could be expected from our experience with the 2D canonical example but also, more importantly, in terms of the resolution of lateral dips (k y ). We considered using the aperture angle θ which, from our 2D experience, could significantly improve the resolution in k x . However, it appeared

Patch acquisition

1353

that θ could not improve the k y resolution because it exhibits the same limitations as |q| in terms of lateral dip resolution. We noted that an ideal ξ parameter should provide the most uniform Tx0 (ξ, k) series when ξ varies, in such a way that the normalized stack Tx0 (k), does not significantly alter the resolution of the final image. Moreover, it must favor among the individual truncation filters Tx0 (ξ, k) those having the best resolution. We choose for ξ the cross product of the inline components of the source and receiver slowness vectors ps and pr (Figure 1):

Ã

ξ =0=

psx ps z

!

Ã

×

pr x prz

!

,

where × denotes the cross product. This parameter involves a type of projection onto the inline plane. For example, the trace gather associated with 0 = 0 contains zero offset traces, but also traces collected at receivers located out of the shot line. The resolution Tx0 (0, k) for 0 = 0 is thus improved in the k y direction and, consequently, the global truncation filter Rx0 (k) is also improved. Moreover, 0 is approximately proportional to the sine of the aperture angle θ and thus tends to favor the contribution of short-offset data which have the best resolution and which are better modeled by the Born approximation. The resolution obtained using ξ = 0 is now studied for the same 3D canonical case as for ξ = |q| (Figure 8). The global truncation filter Rx0 (k) is now greatly improved, both on the k y = 0 and on the k x = 0 sections (compare Figures 6 and 8). For the k y = 0 section, an improvement is obtained not only with respect to the ξ = |q| formula in three dimensions, but also

20km

11.375km

Y

2150m z 50m

Z

(0,0,0)

300m

x

20km

120 receivers

5950 m

50m

★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★

50m

95 shots 4700m ★: Shots

: receivers

(28)

6 swaths

FIG. 4. Acquisition geometry for a swath classical data set. The square on the upper right is a schematic top view of the location of several swaths which have no overlapping lines.

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Lambare´ et al.

to the ξ = θ formula used in two dimensions for the marinetype acquisition [because 0, which is proportional to sin(θ), puts more weight on the short offset traces]. A reduction of the acquisition footprints can also be forecast (Figure 9) on examination of the k y = 0 and k x = 0 sections of the resolution filter Rx0 (k) for a point of type 3 in Figure 5. The sections of the resolution filter are more uniform over 0 [compare Figures 8 and 9 and their counterparts obtained using ξ = |q| (Figures 6 and 7)]. Patch n

S

R

S

a) -10

R

fx (km-1) 0

b) 10

-10

-10

R

5 ●

0

-10

b) 10

-10

1.0

fy (km-1) 0

10

0

Resolution

FIG. 7. (a) k y = 0 and (b) k x = 0 sections of the resolution filter Rx0 (k) for ξ = |q| and for a midpoint of type 3 (see Figure 5). Note the variation of the resolution filters from one midpoint to the next (compare Figures 6 and 7). (a)

(b) -10

fx (km-1) 0

10

-10

fy (km-1) 0

10

0

-10

fz (km-1)

fz (km-1)

-10

0

10

0

0.5

10

1.0 Resolution

Resolution

FIG. 8. (a) k y = 0 and (b) k x = 0 sections of the resolution filter Rx0 (k) for ξ = 0 and for a midpoint of type 5 (see Figure 5). Note the improvement of the resolution both in the k y = 0 and k x = 0 sections compared to those of Figure 6.

-10

-10

b) 10

-10

-10

10

1.0

Resolution

fx (km-1) 0

fy (km-1) 0

10

0

0.5

Resolution

FIG. 6. (a) k y = 0 and (b) k x = 0 sections of the resolution filter Rx0 (k) for ξ = |q| and for a midpoint of type 5 (see Figure 5). Black and white correspond to the optimal and null resolutions, respectively. The lateral resolution in the k x = 0 section is drastically limited.

-10

0

10

scale

10

0

fz (km-1)

0

scale

fz (km-1)

a)

fz (km-1)

-10

fz (km-1)

0.5

10

Resolution

FIG. 5. Nonuniform distribution of large offsets with midpoint. The figure represents a schematic top view of the acquisition geometry for two adjacent swaths. The source and receivers involved in each swath are plotted in black (patch n) and in gray (patch n + 1), respectively. The thick lines represent the source spread and the thin lines represent the perimeter delineated by the receiver layout. The white circles represent five midpoints. The black rectangles represent the area in the inline direction occupied by the sources and receivers during the acquisition associated with one midpoint. These rectangles are split into two parts by the dotted line to indicate that the source maybe to the left or to the right of the midpoint. Two properties of the acquisition are illustrated: (1) the maximum offset coverage varies from one midpoint to the next; (2) the acquisition is asymmetric since the offset coverage for a given midpoint depends on the position of the source relative to the midpoint (except for midpoints of type 1 and 5). fx (km-1) 0

0

S 10

a)

0

S R

R

R

10

-10

: spread of sources : spread of receivers R : contribution area of source-receiver pairs ❍ : CMP S

S

fy (km-1) 0

S

4 ●

S

(31)

scale

R

¯ ¯ 0¯ ¯ ¯. ¯ q¯¯

The term J0 was obtained by factorization of ω2 in the Jacobian |∂(0, k)/∂(LSR, ω)| (Appendix A).

S R

3 ●

(30)

scale

S

1 J0 (r, x, s)1LSR (2π )2 A(r, x, s)

¯ ¯ ∂(0) ¯ ¯ ∂(LSR) J0 (r, x0 , s) = ¯¯ ¯ ∂(q) ¯ ∂(LSR)

R

2 ●

S

E0 (r, x, s) =

fz (km-1)

R

1 X E0 (r, x, s)δG(r, s, T (r, x, s)), (29) [0]max min LSR

with

fz (km-1)

R

δ`(x) =

(1LSR being the increment in trace indexation), and where

Patch n+1

1 ●

S

With ξ = 0, the final 3D ray+Born migration/inversion formula becomes

0

0.5

10

1.0 Resolution

Resolution

FIG. 9. (a) k y = 0 and (b) k x = 0 sections of the resolution filter Rx0 (k) for ξ = 0 and for a midpoint of type 3 (see Figure 5).

3D Ray+Born Inversion—Part 1

The procedure to compute J0 taking advantage of the parameters returned by the 3D paraxial ray tracing is described in Appendix A. CONCLUSION

In this paper, we clarified how ray+Born migration/inversion can handle in different ways the data redundancy which characterizes multichannel acquisitions. Our asymptotic analysis of the Hessian operator (which represents the resolution operator) shows that the final migrated image is built by summation over self-consistent individual migrated images of different resolutions (the migration stack). Each individual migrated images is obtained by migrating a subdata set extracted from the full multichannel data set. The resolution of the migration stack depends of the domain chosen for the CIGs. We identified three criteria to obtain the best migration stack. The first one is to choose a domain for CIGs which insures the maximum resolution of each individual migrated image. Second, the accuracy of the amplitude estimation (i.e., the amplitude of the model perturbations in the frame of a linearized inverse problem) will be improved if the summation is performed over individual migrated images of resolution as uniform as possible. Third, since the resolution of each individual migrated image cannot be strictly equal, the decomposition must be chosen such that the individual migrated images of good resolution are favored in the summation. It is clear that both resolution and signal-to-noise ratio could be improved if a better method of averaging was applied. This would require for each dip to average only the contributions which are effectively significant, which in turn requires an a priori knowledge of the reflectors dips in the image. The ray+Born formalism offers a flexible tool to choose among different domains for CIGs. Taking advantage of this flexibility, we proposed a theoretical guideline based on the analysis of the resolution filter to select the domain for CIGs which allows to verify as much as possible the previous three criteria. For example, we proposed in this paper a migration formula adapted to a 3D swath acquition which is characterized by a poor coverage of lateral dips. These theoretical concepts are verified in our companion paper in the frame of the SEG/EAGE 3D Overthrust experiment (Operto et al., 2003). ACKNOWLEDGMENTS

This work has been partly funded by the European Commission in the framework of the THERMIE programme

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(OG/110/95) MARMOUSI 3-D. We thank IFP and CGG for providing us the overthrust model and data set. Stephane ´ Operto is pleased to acknowledge G. Spence, who contributed substantially to improve this paper. REFERENCES Aminzadeh, F., Brac, J., and Kunz, T., 1997, 3-D salt and overthrust models: Soc. Expl. Geophys. Baina, R., Calandra, H., and Thierry, P., 2001, Demonstration of 3D preserved amplitude PSDM & AVA relevance: 63rd Conf., Eur. Assn. Geosci. Eng., Extended Abstracts, P062. Beydoun, W. B., and Jin, S., 1994, Born or Kirchhoff migration/ inversion: What is the earth’s point of view?: SPIE Proc., Mathematical Methods in Geophysical Imaging II, 82–87. Beylkin, G., 1985, Imaging of discontinuities in the inverse scattering problem by inversion of a causal generalized Radon transform: J. Math. Phys., 26, 99–108. Beylkin, G., and Burridge, R., 1990, Linearized inverse scattering problems in acoustics and elasticity: Wave Motion, 12, 15–52. Bleistein, N., 1987, On the imaging of reflectors in the earth: Geophysics, 52, 931–942. Clochard, V., Nicoletis, L., Svay-Lucas, J., Mendes, M., and Anjos, L., 1997a, Interest of ray Born modelling and imaging for 3-D walkaways: 59th Conf., Eur. Assn. Geosci. Eng., Extended Abstracts, P035. ——— 1997b, 3-D walk-aways imaging in the overthrust model: 59th Conf., Eur. Assn. Geosci. Eng., Extended Abstracts, E047. Forgues, E., 1996, Inversion linearisee ´ multi-parametres ` via la theorie ´ des rais: Ph.D. thesis, Institut Fran¸cais du Petrole–University ´ Paris VII. Jin, S., Madariaga, R., Virieux, J., and Lambare, ´ G., 1992, Twodimensional asymptotic iterative elastic inversion: Geophys. J. Internat., 108, 575–588. Lambare, ´ G., Virieux, J., Madariaga, R., and Jin, S., 1992, Iterative asymptotic inversion in the acoustic approximation: Geophysics, 57, 1138–1154. Lenain, L., Baina, R., Thierry, P., and Calandra, H., 2001, Comparison of kinematic and preserved amplitude PSDM for AVO analysis: 63rd Conf., Eur. Assn. Geosci. Eng., Extended Abstracts, A-07. Lucio, P. S., Lambare, ´ G., and Hanyga, A., 1996, 3D multivalued travel time and amplitude maps: Pageoph, 148, 449–479. Miller, D., Oristaglio, M., and Beylkin, G., 1987, A new slant on seismic imaging: Migration and integral geometry: Geophysics, 52, 943– 964. Operto, S., Lambare, G., Podvin, P., and Thierry, P., 2002, 3D ray+Born migration/inversion—Part 2: Application to the SEG/EAGE overthrust experiment: Geophysics, 68, 1357–1370, this issue. Sevink, A., 1996, Asymptotic seismic inversion: Ph.D. thesis, Technische Universiteit Delft. Tarantola, A., 1986, A strategy for non linear inversion of seismic reflection data: Geophysics, 51, 1893–1903. Thierry, P., Operto, S., and Lambare, ´ G., 1999a, Fast 2-D ray+Born migration/inversion in complex media: Geophysics, 64, 162–181. Thierry, P., Lambare, ´ G., Podvin, P., and Noble, H., 1999b, 3-D preserved amplitude prestack depth migration on a workstation: Geophysics, 64, 222–229. Tura, A., Hanitzsch, C., and Calandra, H., 1998, 3-D AVO migration/ inversion of field data: The Leading Edge, 17, 1578–1583. Xu, S., Chauris, H., Lambare, ´ G., and Noble, M., 2001, Common angle image gather: A strategy for imaging complex media: Geophysics, 66, 1877–1894.

APPENDIX A COMPUTATION OF THE JACOBIAN

The Jacobian of the change of variables (LSR, ω) → (0, k) is given by

¯ ¯ ¯ ∂(L, S, R, ω) ¯ ¯. ¯ J0 = ¯ ∂(0, k) ¯

(A-1)

In case of the swath acquisition, L, S and R denote the index of receiver lines in the swath, the index of shots, and the index

of receivers in one receiver line, respectively. We have

¯ ¯ ¯ ¯ ¯ ∂(L, S, R, ω) ¯ ¯ ∂(0, k) ¯−1 ¯ ¯ ¯ . ¯ = J0 = ¯ ∂(0, k) ¯ ¯ ∂(L, S, R, ω) ¯

(A-2)

Indeed, it is more convenient to compute the terms of the matrix ∂(0, k)/∂(L, S, R, ω), rather than those of ∂(L, S, R, ω)/ ∂(0, k), if we take advantage of the paraxial ray quantities returned by the dynamic ray tracing of Lucio et al. (1996).

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Given that 0 does not depend on frequency, we have

∂0  ∂(L, S, R) ∂(0, k) = ∂q ∂(L, S, R, ω)  ω ∂(L, S, R)

where ps and pr are the source and receiver slowness vectors defined in Figure 1. We have





0

 , 

(A-3)

q

where q is the slowness vector defined in Figure 1. Given the form of the matrix in equation (A-3), it is obvious that ω2 can be factored in the computation of the determinant:

¯ ∂0 ¯ ¯ ¯ ¯ ¯ ∂(0, k) ¯ ¯ ∂(L, S, R) 2¯ ¯ ¯ ¯ ∂(L, S, R, ω) ¯ = ω ¯ ∂q ¯ ¯ ∂(L, S, R)

¯ ¯ 0¯ ¯ ¯. ¯ q¯¯

(A-4)

For numerical implementation, we decomposed the Jacobian as

¯ ¯ ¯ ∂(0, k) ∂(ps , pr , ω) ¯¯−1 ¯ J0 = ¯ , ∂(p , p , ω) ∂(L, S, R, ω) ¯ s

r

(A-5)



∂(0, k) ∂(ps , pr , ω)

∂0  ∂ps = ωI

 0 , q

∂0 ∂pr ωI

where I is the 3 × 3 identity matrix, and,



∂ps  ∂S  ∂(ps , pr , ω)  = ∂(L, S, R, ω)  0  0

0

0

∂pr ∂L 0

∂pr ∂R 0

 0   . 0   1

(A-6)

(A-7)

The terms in the matrix [equation (A-6)] can be easily computed considering that 0 = psx prz − psz prx . All the terms in the matrix of equation (A-7) are returned by paraxial ray tracing.