True-amplitude multicomponent migration of elastic wavefields

Kirchhoff depth migration is an imaging process that transf orms reflection seismic data into the depth domain in order to obtain a structural image of the subsurfac e. Mathematically, it can be formulated as a weighted diffraction stack which is related to the Kirchhoff integral representation of the scalar acoustic wave equation and, hence, usually only form ulated for imaging of compressional or, more generally, monotypical waves. In this paper, the scala r approach to Kirchhoff imaging based on zero-order ray theory is extended to handle the full elast ic wavefield recorded with multicomponent receivers by considering the polarization of respecti v wave modes scattered at an interface. The weight functions that remove the effect of geometrical spre ading from the recorded amplitude change for each scattering mode and have thus to be extended for the c ase of an elastic multicomponent migration. The extended weight functions presented here are a lso v lid for converted waves. It is shown, that the method allows to retrieve the full elastic scatteri ng matrix of target reflectors. INTRODUCTION The graphical migration scheme proposed by Hagedoorn (1954 ) can be formulated as a summation over the Green’s function of the medium. This leads to the so-call ed Diffraction Stack Integral (DSI) as a wave-equation related integral representation of the grap hic l image construction technique (Bleistein and Gray, 2001). In the last two decades, several representatio s of the DSI were developed with slightly different approaches to approximate the (unknown) Green’s fu ction of the medium (see, e. g. Bleistein, 1987; Beylkin and Burridge, 1990). These approaches lead to a weighted diffraction stack in which a weight function accounts for the amplitude loss due to geome trical spreading along the propagation path. If other effects on the recorded amplitude can be neglected, this enables the recovery of the angle-dependent reflection coefficient from the migrated image, and is thus ca lled true-amplitude migration. The approach presented by Schleicher et al. (1993) makes exp licit use of zero-order ray theory in order to describe reflections of smoothly-curved first-order disc ontinuities as the part of the wavefield relevant for imaging. This approach can also be extended to the asympt otically inverse process of demigration, which leads to the so-called unified approach theory to seism ic imaging (Hubral et al., 1996; Tygel et al., 1996). In this paper, the work of the latter three authors, which has so far been strictly formulated only for monotypical reflections (i. e., it did not account for the pos sibility of mode conversions at an interface) is extended to include the vectorial properties of the elast ic wavefield, which can also be described by zero-order ray-theory (̌ Cervený, 2001). This approach can then also handle mode-con verted reflections. RAY-THEORETICAL DESCRIPTION OF ELASTIC AMPLITUDES The correct treatment of the vectorial properties of an elas tic wave within the framework of zero-order ray theory requires a formulation of the problem in ray-centere d coordinates. Assuming that the elastic wavefield has been recorded in three Cartesian components on a (fo r simplicity) planar measurement surface, I Annual WIT report 2002 119 can obtain the principle components in ray-centered coordi nates by means of a rotation ( Červený, 2001): ~ U(~ ξ, t) =   USV USH UP   = H   Ux Uy Uz   . (1) H denotes the local rotation matrix from the ray-centered coo rdinate system to the Cartesian measurement coordinate system. The amplitude ~ U(~ ξ, t) stands for the displacement field and subscript P refers to an incoming P-wave that is polarized perpendicular to the wa vefront1 andSV andSH refer to the two principal S wave components within the local tangent plane t o the wavefront. The correct rotation of the recorded seismograms into the ray-centered coordinate sys tem will thus enable a separation of the wavefield into its constituting wave modes. The rotation itself requi res the knowledge of the local emergence angle of a wavefront; a property which is, as we shall see, also necess ary for the calculation of the weight function. SMR ray branch MRR ray branch