A self-adaptive algorithm for choosing reference velocities in the presence of lateral velocity variations

Seismic wave propagation depicted with the perturbation th eory has important and widespread uses in reflection seismology. As we know, in perturba ion theory, wave propagation needs a reference velocity. The closer the reference ve locity is to the true velocity, the more accurate the wave propagation is. However, it is not easy to choose reasonable reference velocities in the presence of severe lateral v locity variations. Assigning a reference velocity value at each spatial point is not compu tationally feasible, because there is a trade-off between the calculation cost and the num ber of reference velocities. We show that the accuracy of seismic wave propagation can be m or easily improved by choosing a set of reasonable reference velocities rather than by optimizing a one-way wave propagator. Therefore, we introduce a self-adaptive a pproach to choose a set of reference velocities for an extrapolation layer, in the presen ce of lateral velocity variations. Through sorting the velocity data an array with increasing v alues, and by setting a threshold average-velocity ratio or velocityvariance ratio, we can choose a set of reasonable reference velocities for wavefield extrapolation. This met hod can also be used for image edge detecting. It is flexible and computationally cost-eff ective. INTRODUCTION Seismic migration is used to image recorded reflection and sc atter events based on wave theoretical approaches, by de-propagating them to their true su bsurface positions. Wave propagation and de-propagation is based on perturbation theory. No wadays, prestack depth migration (PSDM) has become a most effective tool for imaging complex geolog ical structures. Assuming the macro-velocity field is accurate enough, the migrati on operators or wave propagators should be competent for accurately expressing wave propaga tion in media with severe lateral velocity variations. Stoffa et al. (1990) introduced a first-order split-step cor rection, which is accurate for flat reflectors. However, the accuracy of the migration operator for depicting wave propagation decreases as reflectors become steeper and the differences b etween the reference and migration velocities increase. Therefore, Ristow and Ruhl (1994 ) put forward the Fourier finitedifference (FFD) operator, and Huang et al. (1999) and Rouss eau and de Hoop (2001) intro1email: wang@sep.stanford. edu, shan@sep.stanford. edu 333 334 Wang and Shan SEP–120 duced the generalized-screen propagator GSP, to compensat e for high-order terms neglected when imaging steeply dipping reflectors with the Split-Step Fourier SSF operator. The FFD operator deals with the high-order terms in the frequency-s pace domain; the GSP processes them in the frequency-wavenumber domain. The FFD operator r equi es that the reference velocity is lower than the minimum velocity in an extrapolatio n layer, because the sign of the coefficients in the finite-difference equation can not be cha nged at different lateral points in a depth layer. Otherwise, calculation instability will occu r. Therefore, even if a set of suitable reference velocities is given, the FFD operator cannot give a ry good image. Biondi (2002) modified the general FFD operator by introduci ng the interpolation of two wavefields: the first wavefield is obtained by applying the FFD correction, starting from a reference velocity lower than the medium velocity; the second w avefield is obtained by applying the FFD correction starting from a reference velocity highe r t an the medium velocity. In fact, the high-order terms in wave-propagation perturba tion theory are generated by the velocity perturbation between the true velocity and the ref rence velocity. If the reference velocity is chosen as close as possible to the true velocity, the SSF, FFD and GSP operators can accurately characterize the wave propagation in media with severe lateral velocity variations. Futhermore, the imaging quality can be improved with such pr opagators. Assuming that the macro-velocity model for imaging is accur ate enough, the main reason for most imaging errors is that the one-way wave equation can not accurately characterize the wave propagation in the case of severe lateral velocity v ariations. In such cases, steeply dipping reflectors cannot be clearly imaged. We can decrease the errors either by using an optimized, but complex propagator, or by choosing a set of re asonable reference velocities and using a simple propagator. We think that the latter is much mo re flexible and cost-effective. Of course, both approaches can be combined together. Clapp (2004) proposed that the reference velocity can be sel ected by a generalized Lioyd method. The basic idea behind Lioyd’s method is to iterative ly improve the quantization of a function by looking at the velocity statistics of each regi on (such as mean, median, and variance), and then changing the boundaries of the regions a t each iteration to find the solution which is optimal based on some criterion. Now, we introduce a self-adaptive strategy for selecting a r easonable reference velocity in the presence of lateral velocity variations. The main ste p include setting a threshold for the ratio between two adjoining reference velocities, sort ing he velocity slice into an array, and detecting the edges of velocity regions using velocity a verages and variances in the different regions. The resulting reference velocity field and t he imaging results produced with it indicate that our approach is correct and effective. Mean while, the method is flexible in use, computationally efficient, and easy to program in eithe r 2D or 3D. The approach can also be used for 2D or 3D image edge detecting. Numerical tests dem enstrate that the method is effective. SEP–120 Choosing reference velocities 335 WAVE PROPAGATOR CHARACTERIZED BY PERTURBATION THEORY The Single-Square-Root ( SSR) wavefield-extrapolation equation in the case of lateral ve locity variation is of the following form (Claerbout, 1985): ∂ P (t ,x, y,z) ∂z = ± [ 1 v2 (x, y,z) − ( ∂t ∂x 2 − ( ∂t ∂y 2 ] 1 2 ∂ P (t ,x, y,z) ∂t . (1) The sign convention for the square root is negative for upcom ing wavefield and positive for downgoing wavefield. We decompose the velocity field into two parts: the background velocity field and the velocity perturbation. In fact, we proce ss the slowness field because the decomposition of the slowness is linear. The slowness field d composition is defined as follows: S(x, y,z) = 1 v (x, y,z) = 1 v (z) +4S(x, y,z) = S0 (z)+4S(x, y,z) , (2) whereS0(z) is the background slowness, v (z) the background velocity and 4S(x, y,z) is the slowness perturbation. We hope that the velocity perturbat ion 4S(x, y,z) is as small as possible. Substituting the slowness-field-decomposition equ ation (2) into the wavefield-depthextrapolation equation (1) and discarding the second-orde r terms of the slowness perturbation yields the following formula: ∂ P (t ,x, y,z) ∂z = ± [ 1 v2 (z) +2S0 (z)4S(x, y,z)− ( ∂t ∂x 2 − ( ∂t ∂y 2 ] 1 2 ∂ P (t ,x, y,z) ∂t . (3) Equation (3) can be rewritten as follows: ∂ P (t ,x, y,z) ∂z = ± 