Explicit Seismic Data Depth Extrapolation by an Improved McClellan ...

Explicit Seismic Data Depth Extrapolation by an Improved McClellan Filter Claudio Bagaini, Ernesto Bonomi and Enrico Pieroni

Abstract | A two{dimensional circularly symmetric lter, which is stable, not{aliased, compact and also suciently accurate was designed to perform explicit downward extrapolation of the seismic wave eld by means of the one{way acoustic wave equation. In a rst step the design of a 1D lter was performed in the wavenumber domain by using Tchebyche approximation, while the lter was applied in the space domain. The lter was then extended to the 2D case with a McClellan transformation to exploit its circular simmetry. This convolutional approach requires a short kernel, designed to satisfy local and global constraints in the wavenumber domain. Compared to existing solutions, this allows an improved accuracy, without increasing the computational complexity.

T

I. Introduction

HE aim of seismic migration is to recover the ecographic structure of the earth's subsurface from the pressure eld recorded at the surface, P (X; Y; Z = 0). For this purpose, we are only interested in elds propagating downward in depth and backward in time [1], described by the one{way wave equation [2] that, inside an homogeneous layer [Z;Z +
Z ] with constant velocity v, can be expressed as @ Pb(kx ;ky ;Z; !) = ik Pb(k ; k ; Z; !); x y z @Z

(1)

W is the non{local Fourier transform of the phase shift operator Wb that must be approximated by a nite-length 2D lter Wa . To construct the extrapolator Wa a suitable strat-

egy must be used to avoid the cumbersome straightforward design of a 2D lter and reduce the numerical cost of the convolution in eq. (4). We rst construct, for each value of the design parameter !=v, a nite-length symmetric 1D lter which is then transformed into a 2D lter having an approximate circular symmetry [4]. The adaptation of this approach, formulated in the space domain, to the case of media with laterally variable velocity, is a well established strategy, based on the interpolation of eq. (4) on a set of reference velocities[5]. II. The design of the 1D filter

The construction of the symmetric, nite-length 1D lter can be formulated as a collection of optimization problems in the kx domain by writing, for each value of !=v, a sequence of Tchebyche polynomial approximations of the phase shift operator Wb : N

    X   Wb a kx ; !v = h0 !v + 2 hn !v Tn (cos kx ); n=1

(5)

Tn (cos kx ) = cos(kx n). Because of the symmetries of where Pb is the pressure eld in the wavenumber{frequency where b , one can restrict the study to positive wavenumbers and W domain, kz is the vertical wavenumber de ned by the disfrequencies. For each problem, the ensemble of complex persion relation: coecients hn (!=v) ; n = 0; 1; :: : ;N; must be computed, typs  2  2  2  ically either by implementing least-squares methods that kx + ky ! 1 ? v
t ; (2) minimize the l2 {norm of the error [6], or by constructkz = v
t !
X
Y ing minimax weighted approximations based on the Rekx ; ky are the dimensionless wavenumbers, ! is the dimen- mez exchange algorithm [7]. Our strategy was instead to sionless time frequency and
t;
X;
Y are the time and construct minimax approximations by the simpler, wellspace discretizations. Eq. (1) has the following solution, conditioned problem of Tchebyche interpolation, minde ning the phase shift algorithm: imizing the l1 {norm of the error approximation. To derive b must Wb a , Tchebyche interpolation theory requires that W   Pb(kx ;ky ; Z +
Z; !) = Wb kx ; ky ; !v Pb(kx ;ky ; Z; !) be sampled on the Tchebiche abscissae, i.e. on the zeros of the polynomial TN +1 , given by ki = (2i + 1)=(N + 1)
=2,   ! (3) so that: Wb kx ;ky ; = exp(ikz
Z ) : v

This discussion can be rephrased in the (X;Y ) domain, where eq. (1) becomes a pseudo{dierential equation [3], whose solution, eq. (3), becomes

N

    X b ki ; ! ; Tn (cos ki ) W hn !v = k T1 k2 v n i=0

(6)

where k Tn k2 = N + 1 if n = 0 and (N + 1)=2 if n  1. Tchebyche interpolation theory guarantees that each coecient  O P (X;Y; Z; !) : (4) hn minimizes the largest deviation of Wb a from Wb . P (X;Y; Z +
Z; !) = W X;Y; !v X;Y To avoid aliasing and improve the accuracy of the extrapolated wave eld, some control on the phase shift operator All the authors work in the Center for Advanced Studies, Research Wb is required. For this purpose, we substitute the \origiand Development in Sardinia (CRS4), in the Geophysical Research Area. E{mails: fep,ernesto,[email protected] nal", ideal, lter Wb with a \target" one, imposing a smooth

decay beyond the passband region to provide a high attenuGb(kx ; ky ) = cos k; (11) ation at the Nyquist frequency. To satisfy this requirement, we obtain the iterative relation de ning the Tchebyche the amplitude A of Wb is usually designed in the stopband recursive scheme, in the space domain: region as an inverted cosine [8]. In order to have a faster amplitude decay, we opted for a gaussian tapering: Fn = 2G Fn?1 ? Fn?2 : 

 A kx ; !v =

(

1  if 0  kx  kxtr   2 tr exp kx??kkxtrx ln  ; if kxtr < kx  .

With this notation, the 3D extrapolating formula can be

(7) written in the (X;Y ) domain in terms of Fn (X;Y; Z; !), iteratively obtained by convolution with the kernel G(X;Y ):

where  is the value of A imposed at the Nyquist frequency and kxtr is the transition wavenumber between the passband and the stopband region, de ned as   kxtr = min 0:85

XZ ; !v

Zt sin M ;

P (X;Y; Z +
Z; !) = h0 P (X;Y; Z; !)+2

N X n=1

hn Fn (X;Y; Z; !): (12)

(8) It is now important to observe that in the 2D case the kernel lter Gb does not have a nite-length inverse Fourier

thus limiting the passband region to 85% of the spatial transform; this is not so in the 1D case. From the computaNyquist, or restricting the propagation within an angle M . tional point of view, this makes the convolutional approach The phase  of Wb has been designed as follows: infeasible. We must then look for some good approximation of Gb which is local in the (X;Y ) domain. 8 q  <  kx ; !v = : 

?


?
! v

Zt 2 ? kx

XZ 2 if 0  kx  kxtr ;

akx2 + bkx + c

if kxtr < kx  :

(9)

The three coecients a;b and c are uniquely computed requiring the continuity conditions of  and 0 (derivative with respect to kx ) at the transient frequency kxtr , and  = 0 at the Nyquist. In gs. 1 and 2 we compare the \original" ideal lter ( lled line), the designed \target" (bold line) and the proposed approximation (dots), for dierent values of !=v. Note that both least square and (weighted) minimax approximations of the phase shift operator Wb yield a lter Wb a whose amplitude is not necessarily bound by one. This may be a source of numerical instability during the extrapolation. Our solution consists in normalizing Wb a , for each value of !, dividing by its maximum amplitude on the interval 0  kx  . The only drawback of this approach, as can be observed in g. 2, is a small attenuation of the high frequency components but, since the power spectrum of the seismic data is band limited with a cuto frequency far below the temporal Nyquist, the useful signal is not reduced.

IV. The design of the convolutional kernel

A good candidate is the following form: Gba (kx ; ky ) =

X ij

b gij cos(ikx )cos(jky )

(13)

in which all coecients bgij must be determined. The optimal design of Gba to minimize either the 1{ or the 2{norm of the error is a cumbersome task. To avoid it altogether, J.H. McClellan suggested a very simple recipe [4], basing the lter design on a compromise between local and global constraints. To illustrate this approach, McClellan designed the following 2D low{pass lter: Gba(M ) (kx ; ky ) = ?1 + 21 (1 + cos kx ) (1 + cos ky )

(14)

Recently, Hale [9] brought this technique to the eld of computational 3D migration, and designed an improved McClellan lter: Gb(aH ) (kx ; ky ) = Gba(M ) ? 2c (1 ? cos2kx ) (1 ? cos 2ky ) :

(15)

The parameter c is properly tuned to match a selected value of k along the bisecting diagonal kx = ky . Hale chose III. The design of the 2D filter kx = ky = =3, for which c ' 0:025519. Notice that, with Assuming a 2D grid of data sampled with
X =
Y , the this choice, Gb(aH ) and Gb have coincident Taylor expansions around kx = ky = 0 only up to the second order. b 2D?p lter Wb becomes
rotationally invariant: W (kx ; ky ;!=v) = 2 2 b W kx + ky ;!=v . Consequently, it can be approximated V. A 2D optimal kernel family with the same Tchebyche polynomial expansion used for Our approach is to assume the more general form of eq. the 1D lter [4]: (13) for a symmetric 55 lter with real coecients gij = gji , N       X ! ! ! i; j = 0;: : :; 2. In this way, we aim to achieve high accuracy b Wa k; v = h0 v + 2 hn v cos(nk); (10) at a reasonable computational cost. Actually, imposing the n=1 following constraints: p where k = kx2 + ky2 . Recalling the fundamental recursive property of the Tchebyche polynomials, cos(nk) =  Gba (kx ; 0) = cos kx , ba and G have identical Taylor expansions up to the  G 2 cos(k)cos(n ? 1)k ? cos(n ? 2)k; if we now set fourth order around kx = ky = 0, ba (; ) = ?1 Fbn (kx ; ky ; Z; !) = cos(nk)
Pb(kx ; ky ; Z; !);  the lter minimum is G

we nd a one{parameter family of kernels with coecients

1  23 ? 72b ; g = b; g = 1 24b ? 23  ; g00 = 48 10 20 48 2 2   1 24b ? 25 : g21 = 12 ? b; g22 = 48 (16) 2 The free parameter b permits to require an additional con-

(17)

2

-1

[5]

0

1

2

Fig. 1. Filter amplitude and phase: !=2 = 48:5 Hz. ω = ωmax* 2/3 1.0 0.8 0.6

designed approximated ideal

0.4 0.2 0.0

Fig. 2. Filter amplitude for !=2 = 64:6 Hz. McClellan Hale proposed

0.2

0.0

relative error (%)

absolute error

References 0.2 E. Bonomi, G. Cabitza, Migration of seismic data, CRS4 Internal -0.2 Report, March 1994, Cagliari. J. F. Claerbout, Imaging the Earth's interior, Blackwell Scienti c 0.1 Publications, 1985, pp. 54-62. -0.4 G. Bao and W. W. Symes, Computation of pseudo{dierential operators, SIAM J. SCI. COMPUT. Vol. 17, No. 2, pp.416{429, 0.0 March 1996. -0.6 J. H. McClellan, 1973, The design of two{dimensional digital lters by transformations, Proc. 7th Annual Princeton Conf. on 0.00 0.04 0.08 0.12 0.16 Inform. Sci. and Syst., 573{583. -0.8 C. Bagaini, E. Bonomi, and E. Pieroni, 1994, Split convolutional 0.0 0.2 0.4 0.6 0.8 1.0 approach to 3{D depth extrapolation, 65th Ann. Internat. Mtg, k /π Soc. Expl. Geophys., Expanded Abstracts, 195{198. J. Thorbecke, A. J. Berkhout, 1994, 3{D recursive extrapolation operators: an overview, 64th Ann. Internat. Mtg, Soc. Expl. Fig. 3. Dierences of the three approximated kernels with respect to the exact one in the worst case (kx = ky ). The zoom shows the Geophys., Expanded Abstracts, 1262{1265. relative errors around the origin (dierences are divided by the O. Uzcategui, 1994, Minimax methods to obtain depth migration explicit lters in transversely isotropic media, 64th Ann. Internat. exact kernel). Mtg, Soc. Expl. Geophys., Expanded Abstracts, 1232{1235. G. Blacquiere, 1989, 3{D wave eld extrapolation in seismic depth migration, Delft University of Technology, PhD Thesis. D. Hale, 1991, 3{D depth migration via Mc{Clellan transformations, Geophysics, 56, 1778{1785. x

[6] [7] [8] [9]

3

norm. wavelength

We are very much in debt to our colleague Carlo Maria Nardone for many stimulating discussions and help for organizing the many test runs for our algorithm. The nancial support for this work was provided by the Sardinian Regional Authority.

[4]

1

0

Acknowledgments

[1] [2] [3]

0.4

0.0

phase

which implies a convolution only along the principal axes and the bisecting diagonals. Note that the structure of the kernel proposed in eq. (17) is the same as given by Hale, eq. (15), but our parameter choice results in a fourth order Taylor accuracy compared to the second order accuracy obtained by Hale. Fig. 3 shows the absolute errors, Gb(M ) (kx ; ky ) ? Gb(kx ; ky ), Gb(H ) (kx ;ky ) ? Gb(kx ; ky ) and Gba (kx ; ky ) ? Gb(kx ; ky ), in the worst case, namely along the diagonal kx = ky . The zoom around the origin illustrates the better accuracy of the proposed lter at small wavenumbers. This improved accuracy is particularly meaningful not only because the small wavenumbers are the main components in the seismic signal, but also because to extrapolate seismic data at a typical exploration depth, the complete lter is applied hundreds of times, so that very small dierences in the accuracy produce visible dierences in the acoustic reconstruction of the earth's subsurface.

0.6

0.2

amplitude

1 (1 ? cos2k ) (1 ? cos 2k ) ; Gba (kx ; ky ) = Gba(M ) ? 96 x y

designed approximated ideal

0.8 amplitude

straint, to match speci c necessities in terms of accuracy or computing time. For instance, a smaller computational cost can be achieved setting b = 1=2 in eq. (16). Thus we obtain the following kernel:

ω = ωmax / 2 1.0