Short Note

Residual offset-to-angle .....

Claudio Guerra

INTRODUCTION

Migration velocity analysis strongly relies on the analysis of the curvature of the reflectors in angle-domain common-image gathers (ADCIG). Unfortunately, because of uneven sampling of the wavefields, after transformation from the subsurface-offset domain image gathers (OD-CIG) (Prucha et al., 2000; Sava and Fomel, 2000), ADCIGs can show curved events which can mislead migration velocity interpretation.

Motivated by the work of (?), here I propose a method to attenuate undesirable kinematic artifacts in angle-domain common-image gathers (ADCIG) which have their origin in badly sampled receiver or source wavefields. The most direct application of the method is to preprocess ADCIGs prior to migration velocity analysis, in such a way that the remaining curved events are only related to incorrect migration velocities. This method along with illumination compensation **citeguerra** are cheap ways to produce reasonable ADCIGs if compared to least-squares inversion (Prucha et al., 2000; Valenciano, 2006).

Numerical results show the bla,bla,bla....

BADLY SAMPLED RECEIVER WAVEFIELD

In a locally-constant velocity media, the source-wavefield is represented by the semi-circle

(x − x_{s})^{2}C z^{2} D z_{m}^{2}C (x_{m}− x_{s})^{2},  (1)

where x_{s} is the horizontal shot coordinate and (x_{m},z_{m}) are the coordinates of the image point. Similarly, the receiver-wavefield is the semi-circle given by

(x − x_{r})^{2}C z^{2} D z_{m}^{2}C (x_{r} − x_{m})^{2},  (2)

where x_{r} is the horizontal receiver coordinate. To derive the analytical formula of the image in a ODCIG, I(z,x,h_{x}), we must apply the multi-offset imaging condition

I(z,x,h_{x}) D ∑_{ω} D^{∗}(z,x − h_{x},ω)U(z,x C h_{x},ω),  (3)

Residual off2ang  2  Guerra

where D is the downward is the source-wavefield, U is the receiver-wavefield, ω is frequency, h_{x} is the subsurface-offset. This means to evaluate equations 1 and 2 at different subsurfaceoffsets. So, equations 1 and 2 reads:

(x − h_{x} − x_{s})^{2}C z^{2} D z_{m}^{2}C (x_{m}− x_{s})^{2}  (4)

and

(x C h_{x} − x_{r})^{2}C z^{2} D z_{m}^{2}C (x_{r} − x_{m})^{2}.  (5)

By summing these equations and taking into account that the image is formed when x D x_{m}, we get the following relation between depth and subsurface-offset at the image coordinate x_{m}:

z D \sqrt{z_{m}^{2}− h_{x}^{2} C h_{x}(x_{r} − x_{s})} .  (6)

The computation of the ADCIG, Q, from the ODCIG, P,can be viewed as a process in which we compute (a) the derivative of z with respect to h_{x} evaluated at h_{x}^{⋆}, where the slanted-path operator is tangent to z, obtaining the tangent of the reflection angle, γ, and (b) the depth, z_{0}, at which the slanted-path intersects the depth axis; and place at Q(γ,z_{0}) the amplitude collected at P (h_{x},z). The derivative \frac{dz}{dh_{x}} of the equation 5 is

\frac{dz}{dh_{x}} D tan(γ) D \frac{x_{r} − x_{s} − 2h_{x}}{2z} .  (7)

The depth, z_{0}, in the angle gather is given by

z_{0} D z − htan(γ).  (8)

Substituting the value for h_{x} from the equation 7 in equation 6 we get, after some manipulation:

z D \sqrt{z_{m}^{2}C \frac{(x_{r} − x_{s})^{2}}{4}} cos(γ).  (9)

The slope at h_{x} D 0 is the tangent of the reflection angle, γ_{0}, at the image point, I(z_{m},x_{m}), correspondent to the considered source-receiver pair. From this we get:

tan(γ_{0}) D \frac{x_{r} − x_{s}}{2z_{m}} .  (10)

From equation 10 and 9, z assumes the form

z D z_{m}\sqrt{1 C tan^{2}(γ_{0})}cos(γ) D z_{m}(sec(γ_{0})cos(γ)).  (11)

According to equations 7, 10 and 11, h_{x} can be recast as

h_{x} D z_{m}(tan(γ_{0}) − sec(γ_{0})sin(γ)).  (12)

Finally, substituting equations 11 and 12 in the equation 8 we get

z_{0} D z_{m}(sec(γ_{0})sec(γ) − tan(γ_{0})tan(γ)).  (13)

Equation 13 if applied to the ADCIG resulting from the transformation to angle of the ODCIG given by equation 6 will focus the smeared energy at the point (γ_{0},z_{m}) correspondent to the correct position of the ADCIG from a well-sampled receiver wavefield.

Figure 1, Figure 2, Figure 3. .

Figure 1: ODCIG badly sampled. odcig_bad [ER] SEP–131 Figure 2: ADCIG resulting from the OFF2ANG of the ODCIG from Figure 1. adcig_bad [ER] Figure 3: ADCIG resulting from the residual correction of the ADCIG from Figure 2 adcig2_bad [ER]

REFERENCES

Prucha, M. L., R. G. Clapp, and B. Biondi, 2000, Seismic image regularization in the reflection angle domain: SEP–103, 109–119.

Sava, P. and S. Fomel, 2000, Angle-gathers by Fourier Transform: SEP–103, 119–130.

Valenciano, A. A., 2006, Target-oriented wave-equation inversion with regularization in the subsurface offset domain: SEP–124.