Least-Squares Seismic Migration Using Wavefield Propagators for Avp Inversion
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DRAFT: October 3, 2003 14:01 File: kuehl-sacchi pp.307{329 Page 307 Sheet 1 of 23 CANADIAN APPLIED MATHEMATICS QUARTERLY Volume 10, Number 2, Summer 2002 LEAST-SQUARES SEISMIC MIGRATION USING WAVEFIELD PROPAGATORS FOR AVP INVERSION Based on a presentation at the PIMS{MITACS Workshop on Inverse Problems and Imaging, University of British Columbia, June 9{10, 2001. HENNING KUEHL AND MAURICIO D. SACCHI ABSTRACT. The imaging of geological subsurface struc- tures with seismic reflection energy is a powerful tool for the detection and the assessment of hydrocarbon reservoirs. The seismic method can be understood as a multi-source and multi- receiver reflection scattering experiment. The seismic sources and receivers are placed at the earth's surface. The sources emit seismic energy into the subsurface and the receivers record the earth's response (the backscattered energy) as a function of time and position relative to the source. The goal of seismic imaging is to invert the recorded seismograms for the subsurface structures and their underlying lithological properties. Seismic imaging is usually called migration when only structural image information is desired. The term inversion is used if the goal is to invert for physical properties of the subsurface. In this paper, we do not attempt to invert for true physical param- eters. Instead, we aim at preserving the angle or the closely related ray-parameter dependence of migrated seismic data. In a subsequent step, this information can then be used to invert for the lithological parameters that determine the angle/ray- parameter dependent subsurface reflectivity. In the framework of linearized scattering theory seismic data modelling and mi- gration is defined as a pair of adjoint operators. This definition leads to the formulation of an iterative least-squares seismic migration method. The modelling/migration operator pair is implemented using a wavefield propagator technique. We pro- pose least-squares migration with a regularization term that imposes continuity on amplitude variations as a function of ray-parameter (AVP). This regularization is based on the idea AMS subject classification: 86A15. Copyright c Applied Mathematics Institute, University of Alberta. 307 DRAFT: October 3, 2003 14:01 File: kuehl-sacchi pp.307{329 Page 308 Sheet 2 of 23 308 HENNING KUEHL AND MAURICIO D. SACCHI that roughness along the ray-parameter axis stems from nu- merical artifacts and incompletely and/or irregularly sampled seismic wavefields. A synthetic seismic data example is given for illustration. 1 Introduction. Most common seismic imaging techniques can be subdivided into two categories. The first category are algorithms that back-propagate the seismic surface wavefield into the earth using recur- sive propagators. These algorithms are frequently called wave-equation migration/inversion techniques. The second category are algorithms based on ray-tracing. They are often referred to as Kirchhoff-type imag- ing techniques. In ray-tracing, problems with shadow zones and caustics can occur depending on the complexity of the geology of the subsurface. Furthermore, the ability of the ray-based approach to account for multi- pathing is limited. Propagator techniques, on the other hand, allow the seismic waves to travel along all possible ray-paths (Gray and May, [11]). The formulation of migration/inversion algorithms starts with the definition of a subsurface model in terms of physical parameters and a reflection scattering mechanism. To make the imaging problem tractable we make simplifying assumptions about the model and its interactions with the seismic wavefield. Based on the work of Clayton and Stolt [6], we then derive a forward modelling formula that relates the adopted subsurface model with the seismic data. The ultimate goal in migra- tion/inversion is to solve the forward relationship for the unknown sub- surface model parameters. However, we must keep in mind that, in practice, our simplified model serves merely as a proxy for the more general nature of the problem. We must therefore interpret the results with care. In order to invert the generally \ill-posed" problem we employ a least- squares data fitting method. This allows us to regularize the least- squares solution by imposing certain desirable characteristics and con- straints. Nemeth et al. [20] and Duquet et al. [8] followed the least-squares approach to obtain Kirchhoff-type least-squares imaging techniques. In this paper we investigate the possibility of using wavefield propagators instead. In the first part of this paper, we briefly review seismic data mod- elling based on linear scattering theory. Furthermore, we discuss the use of Green's functions for seismic modelling in spatially varying reference media. This leads to a linear operator equation that relates the subsur- DRAFT: October 3, 2003 14:01 File: kuehl-sacchi pp.307{329 Page 309 Sheet 3 of 23 LEAST-SQUARES SEISMIC MIGRATION 309 face reflectivity with the seismic data. This review is mostly based on a paper by Stolt and Weglein [26]. Next, we introduce a phase-shift propagator method to compute Green's functions in laterally invariant media that is described in Clay- ton and Stolt [6]. The computation of the least-squares inverse requires the knowledge of the adjoint of the modelling operator. The adjoint operator can be regarded as a first approximation to the inverse problem (Claerbout, [5]). In fact, applying the adjoint operator is often sufficient for structural imaging. In this paper we define migration as the adjoint of modelling. Following Stolt and Weglein [26] we describe how to extract amplitude variation with angle (AVA) information from the migrated wavefield. Since propagator techniques are most beneficial when applied to imag- ing problems in complex environments, the modelling/migration opera- tors are generalized for laterally varying media. This can be achieved by expanding the operators used by Clayton and Stolt [6]. The expansion we adopt here results in extended split-step propagators for modelling and migration (Gazdag and Sguazzero, [10]; Stoffa et al., [25]; Kessinger, [13]; Margrave and Ferguson, [18]). As an alternative to AVA imaging/inversion one can use the offset ray-parameter to obtain amplitude variation with ray-parameter (AVP) information (de Bruin et al., [7]; Prucha et al., [23]; Mosher and Foster, [19]). In this paper we will choose the AVP over the AVA parameteri- zation. To complete the discussion on migration using AVP imaging we briefly mention the possibility to improve the adjoint operator by considering the so-called imaging Jacobian (Stolt and Benson, [27]; Sava et al., [24]). The analytical imaging Jacobian can also be used to precondition the iterative least-squares migration discussed in the last part. In least-squares migration we use the above generalized modelling/ migration operators to formulate a least-squares migration/inversion scheme. We introduce a regularization of the inverse problem that appears particularly beneficial when the emphasis is on obtaining am- plitude variation with ray-parameter (AVP) information. An example based on the synthetic Marmousi data set (Versteeg and Grau, [28]) is given for illustration. 2 Seismic data modelling under the Born approximation. Consider the 3-D acoustic wave-equation with variable density ρ(x) and DRAFT: October 3, 2003 14:01 File: kuehl-sacchi pp.307{329 Page 310 Sheet 4 of 23 310 HENNING KUEHL AND MAURICIO D. SACCHI compressional velocity α(x) in the frequency domain: 1 !2 (x; !)Ψ(x; s; !) = + Ψ(x; s; !) D r · ρ(x)r ρ(x)α2(x) (1) = δ(x s); − − where the time-dependence of the wavefield Ψ(x; s; !) is given by e−i!t. The operator (x; !) is the variable density Helmholtz operator. The vector x = (x;Dy; z) denotes the spatial position in a coordinate system with the positive z component pointing down into the earth (Figure 1). The vector s is the position of the point source and the compressional velocity α(x) and density ρ(x) are spatially varying. We define the bulk- modulus K(x) = ρ(x)α2(x) and follow Stolt and Weglein [26] and cast the problem as a perturbation about a reference solution by introducing the reference operator : D0 1 !2 (2) 0(x; !) = + ; D r · ρ0(x)r K0(x) where K0(x) and ρ0(x) are slowly varying local averages of the true values K(x) and ρ(x), too slowly varying to produce significant reflection energy. The scattering operator (scattering potential) (x; !) that generates the seismic reflection data is defined as: V (x; !) = (x; !) (x; !) V D − D0 1 1 1 1 = + !2 (3) r · ρ(x) − ρ (x) r K(x) − K (x) 0 0 a (x) a (x) = 2 + !2 1 ; r · ρ (x) r K (x) 0 0 x x x x where a (x) = ρ0( ) 1 = ∆ρ( ) and a (x) = K0( ) 1 = ∆K( ) . 1 ρ(x) − ρ(x) 2 K(x) − K(x) The coefficients a1(x) and a2(x) are unknown fractional changes of the medium properties describing the subsurface. The differential operators in the term containing a2 cause an angle dependence of the scattering potential. We substitute = + in equation (1) and obtain: D V D0 (4) Ψ(x; s) = Ψ(x; s) δ(x s): D0 −V − − DRAFT: October 3, 2003 14:01 File: kuehl-sacchi pp.307{329 Page 311 Sheet 5 of 23 LEAST-SQUARES SEISMIC MIGRATION 311 z < 0 V(x)=0 x y z > 0 V(x) FIGURE 1: The coordinate system with the z axis pointing into the earth. The scattering potential V is zero for z < 0. The integration boundaries in equation (5) span the entire (model) space. A free surface boundary is not considered. The solution of (4) can be written as an integral equation (Lippman{ Schwinger equation): (5) Ψ(x; s; !) = Ψ (x; s; !) + dx0Ψ (x; x0; !) (x0; !)Ψ(x0; s; !); 0 0 V Z where (6) Ψ (x; s) = δ(x s): D0 0 − − The solution Ψ0(x; s) is the Green's function propagating through the known reference medium.