Geophysical Prospecting, 2011, 59, 256–268

doi: 10.1111/j.1365-2478.2010.00921.x

Preserved-amplitude angle domain migration by shot-receiver wavefield continuation Fr´ed´eric Joncour1 , Gilles Lambar´e2∗ § and Julie Svay-Lucas1 1 Institut

Franc¸ais du P´etrole, Rueil Malmaison, 1-4 Avenue de Bois-Preau, 92852 Rueil-Malmaison, France Now at Total, Avenue Larribau, ´ des Mines de Paris, 1-4 Avenue de Bois-Preau, 92852 Rueil-Malmaison, France 64018 Pau Cedex, France, and 2 Ecole

Received February 2009, revision accepted July 2010

ABSTRACT We present preserved-amplitude downward continuation migration formulas in the aperture angle domain. Our approach is based on shot-receiver wavefield continuation. Since source and receiver points are close to the image point, a local homogeneous reference velocity can be approximated after redatuming. We analyse this approach in the framework of linearized inversion of Kirchhoff and Born approximations. From our analysis, preserved-amplitude Kirchhoff and Born inverse formulas can be derived for the 2D case. They involve slant stacks of filtered subsurface offset domain common image gathers followed by the application of the appropriate weighting factors. For the numerical implementation of these formulas, we develop an algorithm based on the true amplitude version of the one-way paraxial approximation. Finally, we demonstrate the relevance of our approach with a set of applications on synthetic datasets and compare our results with those obtained on the Marmousi model by multi-arrival ray-based preserved-amplitude migration. While results are similar, we observe that our results are less affected by artefacts. Key words: Imaging, Inversion.

INTRODUCTION Preserving amplitude in migration is an important goal for depth imaging. As migration involves both an imaging principle and numerical wave propagation, both aspects have to be considered. If in the past general iterative approaches based on loops of forward and inverse modeling have been investigated (Lailly 1983; Tarantola 1984a,b), most approaches derived for true or preserved-amplitude migration are based on high-frequency asymptotics. In fact high-frequency asymptotics provides us with a powerful physical understanding of depth imaging, which can be used for deriving efficient approximated approaches. This has been the case for ray-based

∗ E-mail: § Now

France

256

[email protected] at CGGVeritas, 1 rue Leon Migaux, 91341 Massy Cedex,

migrations (Beylkin 1984; Bleistein 1987; Jin et al. 1992) but also more recently for one-way wave-equation migration (Zhang, Zhang and Bleistein 2007) with potential application to any type of migration. If true or preserved-amplitude ray-based approaches have been intensively studied in the past (Beydoun and Mendes 1989; Lambar´e et al. 1992; Schleicher, Tygel and Hubral 1993; Thierry, Operto and Lambar´e 1999; Operto, Xu and Lambar´e 2000; Xu et al. 2001), the study of amplitude preservation for migration methods based on numerical resolution of one-way or two-way wave-equations is more recent and connected to the development of these approaches. They are now fully established for depth imaging of complex structures, even if in the case of one-way wave equation the limitations in case of steep dips is a serious drawback that pushes for the use of the two-way wave equation in reverse time migration (RTM) (Mulder and Plessix 2004; Zhang and Sun 2009; Zhang and Zhang 2009; Xu, Zhang and Tang 2010).

 C

2010 European Association of Geoscientists & Engineers

Preserved-amplitude angle domain migration 257

In the present paper we address amplitude preservation for downward continuation migration. We consider the 2D case and a one-way wave equation. In this context an important point is the building of common-image gathers (CIGs) that can be used for migration velocity analysis (Biondi and Symes 2004) or for amplitude versus offset or angle (AVO or AVA) studies. Considering that CIGs in the surface offset domain are definitely not appropriate for wave equation migration due to computing efficiency (Ehinger, Lailly and Marfurt 1996), several other types of gathers have been proposed, such as: ray parameter gathers (de Bruin, Wapenaar and Berkhout 1990; Prucha, Biondi and Fomel 1999; Mosher and Foster 2000; Kuehl and Sacchi 2001, 2002), plane wave parameter gathers (Duquet and Lailly 2006; Soubaras 2006) or angle gathers (Weglein and Stolt 1999; Sava and Fomel 2003; Wu et al. 2004). Concerning this last possibility, ideal for AVA studies, the approach proposed by Sava and Fomel (2003) and derived from the earlier work of Weglein and Stolt (1999) appears particularly appealing because the additional numerical cost for computing the CIGs is very low. In this approach angle domain CIGs (ADCIGs) are computed from slant stacks on individual subsurface offset gathers after the migration. In this paper our main contribution is to introduce amplitude preservation in this approach. Taking advantage of the fact that the images are computed from the locally redatumed wavefield we assume at each image point a locally homogeneous velocity and then use the explicit true amplitude migration formulas that exist for homogeneous reference velocity fields (Cohen and Bleistein 1979; Clayton and Stolt 1981). The strategy for this combination was described in Clayton and Stolt (1981) or Sava, Biondi and Fomel (2001) who used the WKBJ approximation for the redatuming. Compared to these studies but also compared to the work of Zhang et al. (2007), we provide in our paper explicit expressions with demonstrative applications on synthetic data. Finally our paper presents an algorithm for 2D preservedamplitude angle domain migration combining the approaches of Sava and Fomel (2003), for the slant stack on subsurface angle gathers, of Clayton and Stolt (1981) for the preservedamplitude formula and of Zhang, Zhang and Bleistein (2005), for the true amplitude one-way approximation (Cao and Wu 2008, 2010). It is centred on the derivation of a preservedamplitude migration formula, while the numerical scheme for the true amplitude one-way approximation is described in Appendix A. We first review forward Born and Kirchhoff approximations. In the context of a homogeneous reference velocity model, we recall that these linear operators can be ‘diagonalized’ and consequently easily inverted in the spatial

 C

and time Fourier domains. We then demonstrate how these inverse formulas can be introduced in the approach proposed by Sava and Fomel (2003) for computing quantitative ADCIGs. Compared to this pioneering work, our formula mainly differs in the filters applied to the traces and in the weighting factors that are both required for amplitude preservation. Finally we illustrate the relevance of our formulas by applying our new imaging principle to a set of synthetic data sets with curved interfaces and heterogeneous reference velocity models. We also present a comparison with preserved-amplitude ray-Born migration in the complex Marmousi model (Xu 2001; Xu and Lambar´e 2004). ADCIGs obtained with our approach illustrate the advantage of preserved-amplitude shot-receiver migration since they are not affected by migration artefacts observed with the classical ray-based approach (Nolan and Symes 1996, 1997; Stolk and Symes 2004).

PRESERVED-AMPLITUDE MIGRATION PRINCIPLE Forward linearized approximations Preserved-amplitude migration can be addressed in the framework of the linearized inverse problem theory. In this approach, the primarily reflected or diffracted wavefield is expressed in terms of a linear relation of the reflecting or diffracting components of the model. These components can be described by a perturbation of the model parameters (Born approximation) or by a reflectivity distribution (Kirchhoff approximation) depending on the aperture angle at the specific shot and receiver positions. We shall firstly review these two linear approximations. Consider the Green’s function G0 (x, t; s) where s denotes the source position, x the receiver position and t the time for the scalar wave equation (equation (A1)): 

 1 ∂2 2 G0 (x, t; s) = δ(x − s) δ(t). − ∇ c02 (x) ∂t 2

(1)

Let us consider the causal Green’s function, i.e., the solution 0 (x, t; s) = 0 for any t < 0 and such that G0 (x, t; s) = 0 and ∂G ∂t for any position x. In the framework of the Kirchhoff approximation (Bleistein, Cohen and Stockwell 2001), let us consider volumetric reflectivity distribution R (x, θ (r, x, s)) associated to the reflected wavefield, Grefl (r, t; s). Note that the volumetric reflectivity distribution depends on the specific shot, s, and receiver, r, positions through the aperture angle θ (r, x, s). For the first-order Kirchhoff approximation in the time frequency

2010 European Association of Geoscientists & Engineers, Geophysical Prospecting, 59, 256–268

258 F. Joncour, G. Lambar´e and J. Svay-Lucas

two-way traveltime sum of the two one-way traveltimes involved in the Green’s functions, r −→ x and s −→ x and n(x) is the unitary outwards normal to the surface. This substitution is valid within high-frequency asymptotics according to the stationary phase approximation (Bender and Orszag 1978). The reflectivity function accordingly becomes a volumetric reflectivity function. Now if we consider volumetric distribution of squared slowness perturbation, δ(x), associated with the perturbed wavefield, δG(r, ω; s), for the first-order Born approximation (see for example Xu et al. 2001), we find:  δG(r, ω; s) = ω2 dx δ(x)G0 (x, ω; s)G0 (x, ω; r). (6) V

Figure 1 Definition of aperture angle θ and vector q = ps + pr . |q| is the stretching factor.

In the case of a homogeneous reference velocity model c0 (x) = c0 , it is interesting to do Fourier transforms of the reflected/diffracted wavefield according to rx and sx parameters, assuming a 2D configuration (Clayton and Stolt 1981). We define:  +∞ f (kr x ) = dr x f (r x )e−ikr xrx , −∞

1 f (r x ) = 2π

domain, we find: Gr e f l (r, ω; s)  = −iω dx R(x, θ(r, x, s))G0 (x, ω; s)G0 (x, ω; r),

(2)

V

where ω denotes the angular frequency. We use the following convention for the time Fourier transform:  +∞  +∞ 1 f (ω) = dt f (t)eiωt , f (t) = dω f (ω)e−iωt . (3) 2π −∞ −∞ The volumetric reflectivity distribution is related to the distribution of the reflection coefficient, R (x, θ ) through the relation: 2 cos θ (r, x, s) R (x, θ (r, x, s)) . (4) R (x, θ (r, x, s)) = c0 (x) The term 2 cosc θ(r,x,s) is the stretching factor (Fig. 1). Note 0 (x) that the Kirchhoff approximation presented here is derived from the representation theorem (Aki and Richards 1980; Chapman 2004) and high-frequency asymptotic approximations and then extrapolated to general Green’s functions. Note that compared to the expression given by Bleistein et al. (2001) we have extended the surface summation to a volumetric summation and substituted (n (x) .∇x T(r, x, s)) R (x, θ (r, x, s))

(5)

 C

+∞

−∞

dkr x f (kr x )eikr xrx .

(7)

and similarly for sx . Note that we have inverted the sign convention when compare to the Fourier transform in time (equation (3)). For the homogeneous case the analytical expressions for the Fourier transform of the Green’s function are as follows: G0 (x, z, ω; ksx , sz ) =

1 −i(ksx x+ksz (z−sz )) e 2iksz

(8)

with  ksz = ω

1 k2 − sx2 . 2 ω c0

(9)

and similarly for G0 (x, z, ω; krx , rz ) (see Clayton and Stolt 1981). Let us first introduce these expressions in the Born approximation (equation (6)). We obtain: δG(kr x , r z , ω; ksx , sz )  = ω2 dx δ(x)G0 (x, ω; ksx , sz )G0 (x, ω; kr x , r z ) 

V

= ω2

dx V

by its expression at the stationary phase point (right term of equation (4)), i.e., where n is parallel to ∇T. T(r, x, s) is the



=ω

2 δ(ks

δ(x) −i((ks +kr )·x−ksz sz −kr z rz ) e −4ksz kr z

+ kr ) i(ksz sz +kr z rz ) e , −4ksz kr z

(10)

2010 European Association of Geoscientists & Engineers, Geophysical Prospecting, 59, 256–268

Preserved-amplitude angle domain migration 259

where ⎛ ⎜ kr = ⎝

kr x kr z

⎞

⎛

⎟ ⎠

and ks = ⎝

ksx ksz

⎞ ⎠.

(11)

We see that in equation (10) all summations have disapeared. In this sense it is a diagonalized version of the Born approximation, which can be easily inverted using an inverse Fourier transform (Cohen and Bleistein 1979; Clayton and Stolt 1981). In the case of a homogeneous reference velocity model the forward Kirchhoff approximation (equation (2)) can also be diagonalized. This becomes slightly more complicated as the reflectivity distribution depends on the shot and receiver positions, which was not the case for the squared slowness perturbation δ(x) in the Born approximation (equation (6)). The simplification may be introduced in the framework of highfrequency asymptotics again using the stationary phase approximation. Note that this is not an additional approximation since the high-frequency asymptotic approximation had already been introduced for deriving the Kirchhoff approximation (equation (2)). Let us now take a look at the derivation, taking spatial Fourier transform in rx and sx for equation (2). We find:   Gr e f l (kr x , r z , ω; ksx , sz ) = − dsx dr x iω  dx R (x, θ (r, x, s)) G0 (x, ω; s) × V

× G0 (x, ω; r)e−i(ksx sx +kr xrx ) .

(12)

Consider the integrals in sx and in rx . The high-frequency asymptotic Green’s functions are of the form: ⎧ ⎨ G0 (sx , ω) = As (sx )S(ω)eiωTs (sx ) (13) ⎩ G (r , ω) = A (r )S(ω)eiωTr (rx ) 0 x r x √ with S(ω) = 1/ −iω in 2D and where, for simplification, I only expressed the dependencies in (sx , ω) of G0 . Consequently the integral in sx is of the form  dsx R(sx )As (sx )eiωTs (sx ) e−iωpsx sx , (14) where psx = ksx /ω and where, again for simplification, we only expressed the dependencies in (sx ) of R, As and Ts . This integral can be approximated for high-frequency by using the stationary phase approximation. Only the neighbourhood of the stationary points, sspec x , such that ∂ Ts /∂ sx = psx , contributes. In the case of homogeneous Green’s functions, a single stationary point exists (except for grazing incidence). Then

 C

we can write:  dsx R(sx )As (sx )eiωTs (sx ) e−iωpsx sx    ≈ dsx R sxspec As (sx )eiωTs (sx ) e−iωpsx sx ,

(15)

(x, psx , sz ) denotes the x-location of the specular where sspec x source point. It can be obtained by simple geometrical considerations considering straight ray x −→ s in the homogeneous velocity model c0 . We can proceed similarly with the integral in rx . The aperture angle θ can then also be obtained from simple geometrical considerations through equation: tan θ = −

ksx − kr x , ksz + kr z

(16)

where it appears that it only depends on (ksx , kr x , ω). We obtain then for the dependencies of the volumetric reflectivity distribution in expression (12) R(x, θ(ksx , kr x , ω)), which no longer depends on (sx , rx ). We obtain: Gr e f l (kr x , r z , ω; ksx , sz )  = −iω dx R(x, θ(ksx , kr x , ω))G0 (x, ω; ksx , sz ) V

× G0 (x, ω; kr x , r z )  iω dx R(x, θ(ksx , kr x , ω))e−i((ks +kr ).x−ksz sz −kr z rz ) = 4ksz kr z V iω (17) = R(ks + kr , θ(ksx , kr x , ω))ei(ksz sz +kr z rz ) , 4ksz kr z which is a diagonalized expression for the forward Kirchhoff approximation (again in the sense that all summations have disappeared) valid in the case of a homogenous reference velocity model. Symbol . denotes for scalar product.

Common angle imaging Consider an acquisition at sz = 0 and rz = 0. Introducing the distribution of the reflection coefficient rather than the distribution of the reflectivity (equation (4)), we can then simplify the notations in the Kirchhoff approximation (17) to: Gr e f l (kr x , ω; ksx ) =

iω|q| R (ks + kr , θ) −4ksz kr z

(18)

where θ (ksx , kr x , ω) is given by expression (16), q = ps + θ p r , and consequently |q| = 2 cos . The quantitative backward c0 extrapolation of the wavefield from sz = rz = 0 to z is simply: Gback(kr x , ω; ksx , z) = Gr e f l (kr x , ω; ksx )ei(ksz +kr z )z

(19)

since we are in the plane wave domain. We introduce half offset h and midpoint m coordinates and the associated Fourier

2010 European Association of Geoscientists & Engineers, Geophysical Prospecting, 59, 256–268

260 F. Joncour, G. Lambar´e and J. Svay-Lucas

variables (kh , km ). We obtain: ⎧ ⎧ ⎨ kh = ksx − kr x ⎨ h = (sx − r x )/2 and ⎩k = k + k ⎩ m = (s + r )/2 x x m sx rx ⎧ ⎨ khz = ksz − kr z and . ⎩k = k + k mz sz rz

we obtain for the Jacobian:      ∂kmz   ω 4kmz      =  ∂ω   c2 k2 − k2  , kh km hz 0 mz and finally the relation: (20)

We write the data, Grefl (rx , ω; sx ) and the back-propagated data, Gback (rx , ω; sx , z), in this new coordinate system: ⎧ ⎪ ⎨ Gr e f l (h, m, ω) = Gr e f l (m − h, ω; m + h), (21) ⎪ ⎩ Gback(h, m, ω; z) = Gback(m − h, ω; m + h; z) which provides in the spatial Fourier domain: ⎧   km + kh km − kh 1 ⎪ ⎪ , ω; ⎪ Gr e f l (kh , km, ω) = Gr e f l ⎨ 2 2 2  .  ⎪ km + kh km − kh 1 ⎪ ⎪ , ω; ;z ⎩ Gback(kh , km, ω; z) = Gback 2 2 2 (22) By writing the back-propagated wavefield in terms of midpoint and offset coordinates, we directly express the kinematic prestack depth migrated image as, Gback(h = 0, x, t = 0; z) (Claerbout 1985) and the associated subsurface angle gather as Gback(h, x, t = 0; z) (Sava and Fomel 2003). We will now review several properties of the back-propagated wavefield: starting with the expression of equation (19): Gback(kh , km, ω; z) = eikmz z Gr e f l (kh , km, ω),

but also an interesting relation connecting back-propagated Gback(kh , km, ω; z) and Gback(kh , km, t = 0; kz ), to be used in our later derivations. Indeed we know that:

(24)

Because of the Dirac function, the integral in ω will only contribute for the point kz = kmz (kh , km , ω) and will be equal     to Gr e f l (kh , km, ω)/  ∂k∂ωmz . The Jacobian  ∂k∂ωmz  is taken for the value of ω such that kz = kmz (kh , km , ω). Considering: 

kmz

 1 1 (km + kh )2 (km − kh )2 =ω 2 − +ω 2 − , 2 4ω 4ω2 c0 c0

 C

  2 2 2   c0 kmz − khz  Gr e f l (kh , km, ω) ,  Gback(kh , km, t = 0; kmz ) =  ω 4kmz  (27) where kmz and ω are connected through relation (25). Let us go back to the expression of the reflection coefficient. From expressions (18) and (22) we obtain:   2 2 kmz − khz Gr e f l (kh , km, ω) with kh = −tan θ kmz . R(km , θ) = 2iω|q| (28) From this expression, it is possible to obtain the expression of the reflection coefficient R (x, θ) using inverse Fourier transforms in space:  1 dkm R (km , θ) eikm ·x R (x, θ) = 2 (2π )   2 2 −(kmz − khz ) 1 δ (kh + tan θ kmz ) = dk dkh m 2 (2π ) iω|q| × Gr e f l (kh , km, ω)eikm ·x .

(25)

(29)

Let us now introduce relation (27) in the integral expression (29). We obtain:   i2ωkmz R (x, θ) = dkm dkh 2 2 δ (kh + kmz tan θ) π c0 |q|

(23)

Gback(kh , km, t = 0; kz )   1 = dω dz Gback(kh , km, ω; z)e−ikz z 2π   1 = dω dz Gr e f l (kh , km, ω)e−i(kz −kmz )z 2π  = dω Gr e f l (kh , km, ω)δ (kz − kmz (ω, kh , km)) .

(26)

 =

× Gback(kh , km, t = 0; kmz )eikm ·x  i4(ksz + kr z ) δ (kh + kmz tan θ) dkmz dkh π c0 cos θ × Gback(kh , x, t = 0; kmz )eikmz z ,

(30)

where we have used relation |q| = 2 cos θ/c0 at the zero of the Dirac. We can recognize the expression of a Radon transform in the Fourier domain. The Radon transform, F(a, b), of a function f (h, z) is expressed in the spacial or spacial Fourier domain as slant stacks   +∞ dhdz f (h, z)δ(z − a − bh) F (a, b) = −∞

1 = 2π



+∞

−∞

dkh dkmz f (kh , kmz )δ(kh + kmz b)eikmz a . (31)

We recognize in expression (30) a Radon transform over the subsurface angle gathers. In fact, we find:  ∂ 4 dh Gback(h, x, t = 0; z + h tan θ ). R(x, θ) = c0 cos θ ∂z (32)

2010 European Association of Geoscientists & Engineers, Geophysical Prospecting, 59, 256–268

Preserved-amplitude angle domain migration 261

the keys for preserved-amplitude migration as demonstrated in the following examples. We have obtained a preserved-amplitude migration formula for the Kirchhoff approach. In a very similar way, a preservedamplitude formula can be derived for the Born approach starting from expression (10). Assuming that the perturbed wavefield δG is equal to the reflected wavefield Grefl , we compare the Born (equation (10)) and Kirchhoff forward modelling formulas (equation (17)), to find that: δ (ks + kr ) =

ksx − kr x 1 |q|R(ks + kr , θ ) with tan θ = − . iω ksz + kr z (33)

From this relation we can derive a preserved-amplitude Born migration formula:  8 ∂ G (h, x, t = 0; z + h tan θ ), (34) dh δθ (x) = − 2 ∂z back c0 where G back(t) denotes the time integral function of Gback(t), i.e., G back(ω) = (1/iω) Gback(ω). Expressions (34) and (32) are very similar. However, the Born approach for the subsurface angle gather has to be computed with the time integral of the back-propagated data, which cannot be done in the second step after building the subsurface angle gathers. Note that for both formulas (34) and (32), the additional numerical cost for a preserved-amplitude imaging principle is extremely limited. It is done after the full prestack migration and requires saving the subsurface angle gathers. Note that the subsurface angle gathers are processed independently, which fully validates our locally homogeneous approximation.

APPLICATIONS We will now illustrate the relevance of our quantitative migration principle and our paraxial approximation of the true

 C

0

0.

Amplitude 0.5

1.

4 8 Frequency (Hz)

Finally, we propose the following preserved-amplitude common angle wavefield continuation migration algorithm: r Downward extrapolate a wavefield using a true amplitude wave equation propagator (see Appendix A); r Take the zero time downward extrapolated wavefield; Gback (x, h, z + h tan θ ; t = 0) are the subsurface angle gathers in the subsurface offset domain (Sava and Fomel 2003); r Compute the vertical derivative of the subsurface angle gathers; r Apply a weighted slant stack according to equation (32). Our formula turns out to be very similar to the one obtained by Sava and Fomel (2003). The main differences result in the weight |1q| and the depth derivative of the image, which are

12 16 20 24 28

Figure 2 Spectrum of the source used to model the synthetic data set. The source signature is a zero phase impulse band-path filtered by a Butterworth.

amplitude one-way wave equation (Appendix A). To independently assess the amplitude preservation in the imaging principle and redatuming, we first consider the canonical case of a homogeneous reference velocity model. We then consider the case of curved reflectors with a fully heterogeneous reference velocity model. This second example allows us to illustrate the relevance of our two approximations. Finally, we apply our approach to the computation of ADCIGs in the Marmousi model using our preserved-amplitude Born inverse formula. We pursue the possibility of comparing our results with those obtained on the same dataset using a multi-arrivals preserved-amplitude ray-Born migration (Xu et al. 2001; Xu and Lambar´e 2004). For the following applications the source signature used is the transfer function of a Butterworth filter with a frequency bandwidth around 10−20 Hz plotted on Fig. 2.

Canonical case In this case, we use a homogeneous reference velocity model with c0 = 2 km s−1 . We have a flat reflector at 0.5 km depth. The acquisition geometry involves 128 shot and receiver points located at the surface with a 25 m spacing and starting at x = 0 km. 2D data are computed using the analytical formula provided by the ray theory. The reflection coefficient is unitary, i.e., equal to 1 for all incidence angles for all points along the reflector. Figure 3 (left panel) shows the subsurface angle gather obtained for x = 1.5 km. Figure 3 (middle panel) shows the corresponding ADCIG computed with our preserved-amplitude

2010 European Association of Geoscientists & Engineers, Geophysical Prospecting, 59, 256–268

262 F. Joncour, G. Lambar´e and J. Svay-Lucas

0

-500

Offset (m) 0

500

0

0

Angle (degres) 20 40

60

0

Amplitude 0.5

1.0

200

Depth (m)

Depth (m)

200

Vertical wavenumber (1/m)

60°

400

600

400

600

10

20

0° Figure 3 Canonical case: CIGs. Left) subsurface angle gather; middle) ADCIG obtained with our preserved-amplitude migration formula; right) superposition of the spectrum of the various common angle profile of the ADCIG. We show position x = 1.5 km.

|q| =

2 cos θ . c0

(35)

Model with curved reflectors After the first canonical case, our approach is tested in a more complex case composed of curved reflectors in a heterogeneous reference velocity model. For this purpose we used the

 C

0

4.2 km

10 km

7000

5.0

Velocity (m/s)

0 Depth (km)

migration formula (32). We can observe the stretching of the reflectivity profile when the incidence angle increases (Lambar´e et al. 2003). The frequency content of the depth migrated image is shifted, which means that with a bandlimited source signature, we do not investigate the same part of the depth spectrum of the reflectivity function according to the incidence angle. In this context, a correct way to assess the quality of our preserved-amplitude migration formula is to look at the vertical spectrum of the various common angle profiles in Fig. 3 (middle panel). Figure 3 (right panel) shows both the shift in frequency of the reflectivity profiles as well as the spectra with the same shape as the source signature (Fig. 2). All spectra go to the value 1 demonstrating amplitude preservation. Note that if the amplitude preservation is assessed by picking the maximum amplitude along the angle on the ADCIG, then the stretching of the signature has to be compensated by dividing the value by the stretching factor (Bleistein 1987),

2000

Figure 4 Curved reflector case: the model. The model consists of three curved reflectors with a unitary reflectivity (i.e., reflection coefficient equal to 1 for all incidence angles), superimposed on a reference velocity model with a tilted constant gradient of the velocity c0 (x, z) = 1.5 + 0.36x + 0.36z km−1 .

model shown in Fig. 4, where three unitary curved reflectors are superimposed to a reference velocity model with a tilted constant gradient of the velocity c0 (x, z) = 1.5 + 0.36x + 0.36z km−1 . A prestack dataset was computed using a forward ray-Kirchhoff approximation (Baina, Thierry and Calandra 2002) (Fig. 5) and the same source function as for the previous example. Figure 6 shows the subsurface angle gather and ADCIG obtained for x = 5.2 km with our preserved-amplitude scheme. The analysis in the spectral domain of the amplitude preservation along the reflectors would require us to take into account the dip of the event. With synthetic data and well separated events, amplitude preservation can be investigated on AVA curves. These curves are obtained by picking the maximum

2010 European Association of Geoscientists & Engineers, Geophysical Prospecting, 59, 256–268

Preserved-amplitude angle domain migration 263

Receiver position (km) 5.

30

upper reflector middle reflector lower reflector

10. Amplitude

0. 1.

Time (s)

2.

15

3. 0 -60

-40

4. Figure 5 Curved reflector case: the reflected wavefield. Common shot gather for xs = 5.km computed using ray-Kirchhoff modelling and the same source function as the previous example.

0.0

redatumed offset (km) Depth Offset (m) .0 .2 400.4 -400 -200 0 200

60. -60

0.0

aperture angle (degree) Common angle gather (degree) .0 60. -40 30. -20 0 20 30.4 0

1.

3.

3000

Depth (km) Depth (m)

Depth (km) Depth (m)

2.

2000

1000

2.

2000

3.

3000

4000

4.

4000

5.

5000

5000

4. 5.

Figure 6 Curved reflector case: CIGs. Left) subsurface angle gather at x = 5.3 km computed with our true amplitude one-way paraxial extrapolator; right) ADCIG at x = 5.3 km computed with our preserved-amplitude migration principle.

amplitude on ADCIG for each reflector and then correcting for the stretching factor (equation (35)). Figure 7 provides the AVA curves obtained by picking at x = 4.2 km the maximum amplitude along the ADCIGs obtained with our preserved-amplitude formula after correction by the stretching factor (equation (35)). We obtain flat AVA curves and note that all the curves go to the same level (equal to the maximum source amplitude in time), thus demonstrating the good compensation of amplitudes for propagation issues. This example demonstrates both the accuracy of our preservedamplitude migration principle and our numerical implementation of paraxial approximation of the true amplitude oneway operator proposed by Zhang, Zhang and Bleistein (2003). Figure 8 shows the stacked migrated image obtained stacking the ADGIGs.

 C

60

1.

3.

X (km) 5.

7.

9.

1. Depth (km)

1.

40

Figure 7 Curved reflector case: AVA curves. Preserved-amplitude Kirchhoff migration AVA curves obtained by picking at x = 4.2 km the maximum amplitude along the ADCIGs obtained with our preserved-amplitude formula, after correction for the stretching factor (equation (35)).

0. 1000

-20 0 20 Angle (degrees)

2. 3. 4. 5.

Figure 8 Curved reflector case: Kirchhoff migration stack. Depth migrated image obtained stacking the ADCIGs.

Marmousi model The Marmousi is a well-known synthetic model and data set (Bourgeois et al. 1991) (Fig. 9 (top panel)). It is often used for testing migration in complex media (Versteeg and Grau 1991; Versteeg 1993; Geoltrain and Brac 1993; Ettrich and Gajewski 1996; Audebert et al. 1997). Some results have even been obtained in terms of amplitude preservation (Operto et al. 2000; Xu et al. 2001; Xu and Lambar´e 2004). These results were based on multi-arrival preserved-amplitude rayBorn migration and provided by the end relative impedance perturbation profiles that compared advantageously to the exact profiles (Fig. 9 (bottom panel)). The relative impedance perturbation is defined by ratio δIp /Ip0 where δIp is the perturbation of impedance and Ip0 is the impedance in the reference model. These results were obtained with a smooth version of the exact velocity model as a reference velocity model (Fig. 9 (middle panel)). Note that contrary to the ray-Born approach our migration approach does not require such a smooth reference velocity model. In fact, the migrated result obtained

2010 European Association of Geoscientists & Engineers, Geophysical Prospecting, 59, 256–268

264 F. Joncour, G. Lambar´e and J. Svay-Lucas

Distance (m)

Distance (m)

0

9100

0

Marmousi velocity model

2950 Exact relative impedance perturbation

Distance (m) 0

8000

Depth (m)

Depth (m) 2700

3000 100

5500

velocity (m/s)

0

0

9100

Distance (m)

5500

2700

0

Smooth velocity model

2950

Distance (m)

Migrated stack Section

9100

Depth (m) 2700

0.4

Distance (m) 8000

0 -0.2 -0.4

Relative impedance perturbation model

Figure 9 Marmousi case: the model. Top) the Marmousi velocity model; middle) a smooth version of the previous velocity model that can be used as a reference velocity model for multi-arrival ray-Born migration (Thierry et al. 1999; Operto et al. 2000; Xu et al. 2001; Xu and Lambar´e 2004); bottom) model of relative impedance perturbation obtained from the previous smooth velocity model.

on Marmousi with our approach is improved when using the exact model as the reference velocity model (Fig. 10 (bottom panel)). However, to compare these results with multi-arrival preserved-amplitude ray-Born migration, we will use the same smooth reference velocity model. Figure 10 (middle panel) shows us the result of preservedamplitude prestack migration (the common image stack of the ADCIGs) as compared to the exact relative impedance profile (Fig. 10 (top panel)). This migration stack is not preserved-amplitude; normalization according to the aperture angle range has not been introduced. However we see that the structure and the balance of amplitude have been wellrecovered. Let us now analyse amplitude preservation on ADCIGs, by looking at the subsurface angle gather and ADCIG obtained

 C

3000 100

0.2

Depth (m)

0

impedance pertubation

0

8000

Depth (m)

Depth (m)

velocity (m/s)

3000 100

2950 Migrated stack Section

Figure 10 Marmousi case: preserved-amplitude Born migration. Top) the exact relative impedance perturbation; middle) the preservedamplitude migration stack obtained by stacking the ADCIGs. The reference velocity model is a smooth version of the exact velocity model shown on the Fig. 9 (middle); bottom) the preserved-amplitude migration stack obtained stacking the ADCIGs. The reference velocity model is the exact one as shown in Fig. 9 (top).

with our preserved-amplitude migration formula in the complex part of the Marmousi model, i.e., in x = 6.2 km (Fig. 11). We see that they are quite smooth. When we compare them to the ones obtained with multi-arrival preserved-amplitude rayBorn migration, we see that they are much less noisy (Fig. 12). An interesting point, as predicted by Stolk and Symes (2004), is that they are not affected by migration artefacts created by the violation of the imaging condition and indicated by red arrows on Fig. 12. In fact, these artefacts are due to multipathing when tracing rays from the surface. Due to the redatuming, this problem simply disappears in Claerbout’s migration approach (Claerbout 1985). If we compare this to the ray-Born

2010 European Association of Geoscientists & Engineers, Geophysical Prospecting, 59, 256–268

Preserved-amplitude angle domain migration 265

Figure 13 Marmousi case. Comparison with the exact relative impedance perturbation profile (red) for x = 6.2 km with the one obtained with our preserved-amplitude migration. Aperture angle contributions from 10–30 degrees have been stacked.

Figure 11 Marmousi case: CIGs. Left) subsurface angle gather for x = 6.2 km; right) ADCIG obtained with our preserved-amplitude migration using the smooth reference velocity model (Fig. 9 (middle)).

Figure 12 Marmousi case: comparison with ray-Born approach. Left) ADCIG obtained with our preserved-amplitude migration using the smooth reference velocity model (Fig. 9 (middle)) for x = 6.2 km; right) ADCIG obtained with ray-Born preserved-amplitude migration using the smooth reference velocity model (Fig. 9 (middle)) (Xu et al. 2001). Red arrows on the right panel indicate some migration artefacts created by the violation of the imaging condition (Stolk and Symes 2004).

the absolute amplitude of the source function, it was estimated considering the relative energy of both profiles. CONCLUSIONS We have shown how amplitude preservation could be introduced in 2D angle domain source-receiver migration. Our approach involves the use of both a true amplitude one-way paraxial approximation (Zhang et al. 2003) and a new imaging formula. Similarly to Sava and Fomel (2003) our formula involves a slant stack on subsurface angle gathers. However, there are significant differences that are important for amplitude preservation. The implementation of our migration principle is extremely simple as soon as the true amplitude redatuming has been performed. The extra cost is negligible considering the computing time required by the redatuming step. The relevance of our formula has been demonstrated on several synthetic data sets with increasing complexity. Our work is limited to the 2D acoustic case. The approach proposed by Sava and Fomel (2003) has been extended to the 3D case (Fomel 2004) and even to the anisotropic and converted wave cases (Biondi 2007; Rosales et al. 2008). Even if the extension of our results to these new cases is beyond the scope of this paper, we believe that preserved-amplitude migration formulas could be derived in a similar manner. ACKNOWLEDGMENTS

result we see that the boundaries of the ADCIG are smoothed. It may consequently be difficult to know the range of aperture angles effectively covered by the acquisition. Figure 13 presents a comparison of the relative impedance perturbation profile obtained with our migration approach with the exact impedance profile band-pass filtered according to the source (the vertical profile was converted to vertical time, then filtered according to the source bandwidth and finally back converted to depth). Note that as we did not have

 C

The authors thank Paul Sava and four anonymous reviewers for their help in improving the paper and Reda Baina for the linearized ray-Kirchhoff modelling code used to model the synthetic data. REFERENCES Aki K. and Richards P.G. 1980. Quantitative seismology: Theory and methods. W. H. Freeman & Co.

2010 European Association of Geoscientists & Engineers, Geophysical Prospecting, 59, 256–268

266 F. Joncour, G. Lambar´e and J. Svay-Lucas

Audebert F., Nichols D., Rekdal T., Biondi B., Lumley D. and Urdaneta H. 1997. Imaging complex geologic structure with single-arrival Kirchhoff prestack depth migration. Geophysics 62, 1533–1543. Baina R., Thierry P. and Calandra H. 2002. Demonstration of 3D preserved amplitude PSDM & AVA relevance. The Leading Edge 21, 1237–1241. Bamberger A., Engquist B., Halpern L. and Joly P. 1988a. Higher order paraxial wave equation approximations in heterogeneous media. SIAM Journal on Applied Mathematics 48, 129–154. Bamberger A., Engquist B., Halpern L. and Joly P. 1988b. Paraxial approximations in heterogeneous media. SIAM Journal on Applied Mathematics 48, 99–128. Bender C. and Orszag S. 1978. Advanced Mathematical Methods for Scientists and Engineers. McGraw-Hill. Beydoun W.B. and Mendes M. 1989. Elastic ray-Born 2 – migration/inversion. Geophysical Journal International 97, 151–160. Beylkin G. 1984. The inversion problem and applications of the generalized radon transform. Communications on Pure and Applied Mathematics 37, 579–599. Biondi B. 2007. Angle-domain common-image gathers from anisotropic migration. Geophysics 72, S81–S91. Biondi B. and Symes W. 2004. Angle-domain common-image gathers for migration velocity analysis by wavefield-continuation imaging. Geophysics 69, 1283–1298. Bleistein N. 1987. On the imaging of reflectors in the Earth. Geophysics 52, 931–942. Bleistein N., Cohen J. and Stockwell J. 2001. Mathematics of Multidimensional Seismic Imaging, Migration and Inversion. Springer. Bourgeois A., Bourget M., Lailly P., Poulet M., Ricarte P. and Versteeg R. 1991. Marmousi, model and data: Proceedings of the 1990 EAEG workshop on “Pratical Aspects of Seismic Data Inversion”, Expanded Abstracts, 5–16. Cao J. and Wu R.-S. 2008. Amplitude compensation for one-way wave propagators in inhomogeneous media and its application to seismic imaging. Communications in Computational Physics 3, 203–221. Cao J. and Wu R.-S. 2010. Lateral velocity variation related correction in asymptotic true-amplitude oneway propagators. Geophysical Prospecting 58, 235–243. Chapman C. H. 2004. Fundamentals of Seismic Wave Propagation. Cambridge University Press. Claerbout, J. 1971. Toward a unified theory of reflector mapping. Geophysics 36, 467–481. Claerbout J. F. 1985. Imaging the Earth’s Interior. Blackwell Scientific Publications. Clayton R. W. and Stolt R. 1981. A Born-WKBJ inversion method for acoustic reflection data. Geophysics 46, 1558–1565. Cohen J. K. and Bleistein N. 1979. Velocity inversion procedure for acoustic waves. Geophysics 44, 1077–1087. Collino F. 1987. Analyse num´erique de mod`ele de propagation d’ondes. Application a` la migration et a` l’inversion des donnees sismiques. PhD thesis, Universit´e Paris XI Dauphine. Collino F. 1996. Perfectly Matched Absorbing Layers for the Paraxial Equations, Rapport de Recherche n2964. Technical report, INRIA, Domaine de Voluceau, Rocquencourt, BP 105, 78153 Le Chesnay Cedex (France).

 C

Collino F. and Joly P. 1995. Splitting of operators, alternate directions and paraxial approximations for the 3-D wave equation. SIAM Journal on Scientific Computing 16, 1019–1048. de Bruin C. G. M., Wapenaar C. P. A. and Berkhout A. J. 1990. Angledependent reflectivity by means of prestack migration. Geophysics 55, 1223–1234. Deng F. and McMechan G. 2007. True-amplitude prestack depth migration. Journal of Seismic Exploration 72, S155–S166. Duquet B. 1996. Amelioration de l’imagerie sismique de structures geologiques complexes. PhD thesis, Universit´e de Paris XIII. Duquet B. and Lailly P. 2006. Efficient 3D wave-equation migration using virtual planar sources. Geophysics 71, S185–S197. Ehinger A., Lailly P. and Marfurt K. 1996. Green’s function implementation of common-offset, wave-equation migration. Geophysics 61, 1813–1821. Ettrich N. and Gajewski D. 1996. Wavefront construction in smooth media for prestack depth migration. Pure and Applied Geophysics 148, 481–502. Fomel S. 2004. Theory of 3-D angle gathers in wave-equation imaging. 74th SEG meeting, Denver, Colorado, USA, Expanded Abstracts, 1053–1056. Geoltrain J. and Brae S. 1993. Can we image complex structures with first-arrival travel-time? Geophysics 58, 564–575. Jin S., Madariaga R., Virieux J. and Lambar´e G. 1992. Twodimensional asymptotic iterative elastic inversion. Geophysical Journal International 108, 575–588. Kuehl H. and Sacchi M. 2001. Generalized least-squares DSR migration using a common angle imaging condition. 71st SEG meeting, San Antonio, Texas, USA, Expanded Abstracts, 1025–1028. Kuehl H. and Sacchi M. 2002. Robust AVP estimation using leastsquares wave-equation migration. 72nd SEG meeting, Salt Lake City, Utah, USA, Expanded Abstracts, 281–284. Lailly P. 1983. The seismic inverse problem as a sequence of before stack migrations. Conference on Inverse Scattering, Theory and application, SIAM, Philadelphia, Pennsylvania, USA, Expanded Abstracts, 206–220. Lambar´e G., Operto S., Podvin P. and Thierry P. 2003. 3D ray-Born migration/inversion – Part 1: Theory. Geophysics 68, 1348–1354. Lambar´e G., Virieux J., Madariaga R. and Jin S. 1992. Iterative asymptotic inversion in the acoustic approximation. Geophysics 57, 1138–1154. Mosher C. and Foster D. 2000. Common angle imaging conditions for pre-stack depth migration. 70th SEG meeting, Calgary, Canada, Expanded Abstracts, 830–833. Mulder W. and Plessix R.-E. 2004. A comparison between oneway and two-way wave-equation migration. Geophysics 69, 1491–1504. Nolan C. and Symes W. 1996. Imaging and coherency in complex structures. 76th SEG meeting, New Orleans, Louisiana, USA, Expanded Abstracts, 359–362. Nolan C. and Symes W. 1997. Global solutions of a linearized inverse problem for the wave equation. Communications on Partial Differential Equations 22, 919–952. Operto S., Xu S. and Lambar´e G. 2000. Can we image quantitatively complex models with rays? Geophysics 65, 1223–1238. Prucha M., Biondi B. and Fomel S. 1999. Angle domain common image gathers by wave equation migration. 69th SEG

2010 European Association of Geoscientists & Engineers, Geophysical Prospecting, 59, 256–268

Preserved-amplitude angle domain migration 267

meeting, Houston, Texas, USA, Expanded Abstracts, 824– 827. Rosales D., Fomel S., Biondi B. and Sava P. 2008. Wave-equation angle-domain common-image gathers for converted waves. Geophysics 73, S17–S28. Sava P., Biondi B. and Fomel S. 2001. Amplitude preserved common image gathers by wave equation. 71st SEG meeting, San Antonio, Texas, USA, Expanded Abstracts, 296–299. Sava P. and Fomel S. 2003. Angle-domain common gathers by wavefield continuation methods. Geophysics 63, 1065–1074. Schleicher J., Tygel M. and Hubral P. 1993. 3D true-amplitude finiteoffset migration. Geophysics 58, 1112–1126. Soubaras R. 2006. Modulated-shot migration. 76th SEG meeting, New Orleans, Louisiana, USA, Expanded Abstracts, 2430–2433. Stolk C. and Symes W. 2004. Kinematic artifacts in prestack depth migration. Geophysics 69, 562–575. Tarantola A. 1984a. Inversion of seismic reflection data in the acoustic approximation. Geophysics 49, 1259–1266. Tarantola A. 1984b. Linearized inversion of seismic reflection data. Geophysical Prospecting 32, 998–1015. Thierry P., Operto S. and Lambar´e G. 1999. Fast 2D ray-Born inversion/migration in complex media. Geophysics 64, 162–181. Versteeg R. 1993. Sensitivity of prestack depth migration to the velocity model. Geophysics 58, 873–882. Versteeg R. J. and Grau G. 1991. The Marmousi experience. Proceedings of the 1990 EAEG workshop on Pratical Aspects of Seismic Data Inversion. Weglein A. B. and Stolt R. 1999. Migration-inversion revisited (1999). The Leading Edge 18, 950–953. Wu R.-S., Luo M., Hen S. and Xie X.-B. 2004. Acquisition aperture correction in angle-domain and true-amplitude imaging for wave equation migration. 74th SEG meeting, Denver, Colorado, USA, Expanded Abstracts, 937–940. Xu S. 2001. Migration/inversion en milieu complexe par des m´ethodes ´ asymptotiques haute fr´equence. PhD thesis, Ecole des Mines de Paris. Xu S., Chauris H., Lambar´e G. and Noble M. 2001. Common angle image gather: a strategy for imaging complex media. Geophysics 66, 1877–1894.

Zhang Y., Zhang G. and Bleistein N. 2003. True amplitude wave equation migration arising from true amplitude one-way wave equations. Inverse Problems 19, 1113–1138. Zhang Y., Zhang G. and Bleistein N. 2005. Theory of true amplitude one-way wave equations and true-amplitude common-shot migration. Geophysics 70, E1–E10. Zhang Y., Zhang G. and Bleistein N. 2007. True-amplitude, angledomain, common image gathers from one-way wave equation migrations. Geophysics 72, S49–S58. Zhang Y. and Zhang H. 2009. A stable TTI reverse time migration and its implementation. 79th SEG meeting, Houston, Texas, USA, Expanded Abstracts, 2794–2798.

APPENDIX A: ‘TRUE AMPLITUDE’ ONE WAY PARAXIAL APPROXIMATION Consider the 2D scalar wave equation for the wavefield P(x, t) where x denotes the position, and t the time,   1 ∂2 2 P(x, t) = 0, (A1) − ∇ c02 (x) ∂t 2 and c0 (x) denotes the reference velocity model. There are no source terms in equation (A1), which is only valid for wave propagation away from the source locations. Zhang et al. (2003) proposed that to maintain both kinematics and dynamics of equation (A1), we use the following one-way wave equations for shot and receiver migration: ⎧  ∂ ⎪ ⎪ − i D(x, ω) − D(x, ω) = 0 ⎪ ⎨ ∂z (A2)   ⎪ ⎪ ⎪ ∂ + i U(x, ω) − U(x, ω) = 0 ⎩ ∂z where D(x, ω) and U(x, ω) denote the downgoing and upgoing wavefields, respectively, ω denotes the angular frequency and the  and operators are defined by:

⎧    ⎪ ⎪ ω ∂ 2 1 ⎪ ⎪ (x, ω) = c 1 + (x) ⎪ 0 ⎪ ⎨ c0 (x) ω2 ∂x ⎞ ⎛  2 −1  2  ⎪ ⎪ ∂c (x) 1 ∂ ∂ 0 ⎪ 2 ⎪ ⎠. ⎝1 − ω + c0 (x) c0 (x) (x, ω) = ⎪ ⎪ ⎩ 2c0 (x) ∂z ∂x ∂x Xu S. and Lambar´e G. 2004. Fast migration/inversion with multiarrival ray fields. Part I: Method, validation test and application in 2D to Marmousi. Geophysics 69, 1311–1319. Xu S., Zhang Y. and Tang B. 2010. 3D angle gathers from reverse time migration. 80th SEG meeting, Denver, Colorado, USA, Expanded Abstracts, ID499. Zhang Y. and Sun J. 2009. Practical issues of reverse time migration: true-amplitude gathers, noise removal and harmonic-source encoding. First Break 26, 19–25.

 C

(A3)

Equation (A2) differ from the original one-wayapproximation proposed by Claerbout (1971), by the derivatives of the velocity in term  and more significantly by the term , which cancels in case of a homogeneous reference velocity model. Zhang et al. (2003) demonstrated that when applying high-frequency asymptotics to their new one-way equation both the traveltime and zero-order amplitude terms are the same as those of the ray theory. This is valid for

2010 European Association of Geoscientists & Engineers, Geophysical Prospecting, 59, 256–268

268 F. Joncour, G. Lambar´e and J. Svay-Lucas

∂ heterogeneous velocity models, which is not the case for the P (x, z, ω) = ±i N P (x, z, ω) (A6) ∂z classical one-way equation. Let us first consider the problem of the numerical implewith the initial condition P (x, zi , ω) = P(x, zi , ω). (Positive mentation of equation (A2). Zhang et al. (2005) proposed sign for an up-going wave and negative sign for a down-going the use of a finite-difference scheme based on the paraxial wave) Implicit schemes are prefered to explicit schemes for approximation. their stability even in the case of strong lateral variations. The In our implementation we use a different numerical scheme. term is then solved using an explicit scheme for computing Our scheme is still a finite-difference scheme, but we use a P (x, zi+1 , ω) solution of: splitting approach (Collino and Joly 1995). Let us first review ∂ P (x, z, ω) = P (x, z, ω) (A7) the one way paraxial approximation. We approximate the ∂z operator , using the Pad´e expansion ⎞ ⎛   −1    N  ω ⎝ ∂ 2 ∂ 2⎠ 2 α N,n c0 (x) , (A4) ω + β N,n c0 (x) 1+  N(x, ω) = c0 (x) ∂x ∂x n=1 where β N,n and α N,n are Pad´e approximants that can be determined by several methods (Collino 1987; Bamberger et al. 1988a,b). Here, β N,n and α N,n are taken such as: ⎧   nπ ⎪ 2 ⎪ ⎪ α N,n = cos ⎨ 2π + 1 (A5)   ⎪ 2n π 2 ⎪ ⎪ β N,n = sin2 ⎩ 2N + 1 2N + 1 where N denotes the order of the Pad´e approximation. The aperture angle of the validity domain of paraxial approximation increases as N increases. We propose the following numerical scheme for computing P(x, zi+1 , ω) from P(x, zi , ω). We first use the implicit finite difference scheme proposed by Collino (1987) and implemented by Duquet (1996) or Ehinger et al. (1996) for computing P (x, zi+1 , ω) solution of:

 C

with the initial condition P (x, zi , ω) = P (x, zi+1 , ω). We choose P(x, zi+1 , ω) = P (x, zi+1 , ω).

(A8)

The use of a first-order explicit scheme is not an issue here, because the term essentially involves an amplitude correction whereas the N term addresses the kinematic aspects. perfectly matched absorbing layers are introduced for the paraxial equations (Collino 1996) to avoid spurious reflections at the boundaries. Note that the true amplitude one-way approximation that we consider, only takes into account the amplitude variations due to the geometrical spreading and not those due to transmission loses as considered for example by Deng and McMechan (2007).

2010 European Association of Geoscientists & Engineers, Geophysical Prospecting, 59, 256–268