DSR Migration Velocity Analysis by Differential Semblance Optimization

Transcription

1 DSR Migration Velocity Analysis by Differential Semblance Optimization A. Khoury (Total E&P France), W. W. Symes (Rice University), P. Williamson and P. Shen (Total E&P USA Inc.) Society of Exploration Geophysicists, New Orleans, Oct 6

2 Agenda Automated construction of velocity models based on optimization and prestack imaging can succeed, given appropriate choice of objective function, and kinematically consistent prestack imaging engine Cost: O() migrations. Factors affecting accuracy include () aperture, () coherent noise in image volume originating either in data or in imaging operator.

3 Outline Objective approaches to automated velocity analysis Differential semblance for DSR migration Examples Conclusion

4 Objective approaches to automated velocity analysis 3

5 Objective Velocity Analysis Concept: identify an objective function J[v;d] so that kinematic consistency of data d, velocity v J[v;d] extremal Proposed objectives for velocity estimation from reflection data: Output Least Squares (Tarantola 984) Stack Power / Semblance (Toldi 985) Differential Semblance (Symes 986) All take form J[v; d] = W[v]d ; W[v] is a v-dependent weighting operator. 4

6 Properties of Objectives Size of typical problems must use gradient (Newton-based) optimization. objective must be smooth as function of velocity, data. Theorem: J[v;d] smooth in v,d and quadratic in d differential semblance (Stolk & Symes 3). Numerics: Output Least Squares, Semblance objectives have many spurious local minima, whereas Differential Semblance is (sometimes provably) unimodal. Upshot: Gradient-based optimization for large velocity updates from wideband reflection data Differential Semblance. 5

7 Differential semblance for DSR migration 6

8 Survey Sinking Claerbout: Reflected wavefield extrapolated to depth z = solution p(m, h, z, t) of p z = v (s, z) t + s + v (r, z) t + p r where s = m h is source x, r = m+h is receiver x, and m and h are (subsurface, not survey!) midpoint and half-offset. Initial value at z = is data: p(m, h,, t) = d(m, h,t). Assumes upcoming reflected wavefield, zero phase wavelet,... This paper: D propagation - 3D description similar with common azimuth approximation (Biondi and Palacharla, 996). 7

9 Depth Stepping Operator square roots must be approximated - we used the Generalized Screen Propagator ( GSP ) approach (Wu, 994), following DeHoop and Le Rousseau (). After discretization in depth, extrapolation given by p n+ = H n [v]p n, p = d where H n [v] = GSP approximation to DSR operator, using velocity values in nth depth slab {z n z z n+ }, and p n (m, h, t) = p(m, h,z n,t). 8

10 Imaging and Focusing Claerbout s imaging principle: Image extracted from extrapolated field at zero time, zero offset: I(m, z n ) = p n (m,, ) If velocity model is correct, then energy in zero time field (offset image gather) is focused at zero (subsurface) offset: p n (m, h, ), h [Other possible imaging principles: angle image gathers (Prucha et al 999, Sava and Rickett ), time lag gathers (Sava and Fomel, 5).] 9

11 Differential Semblance for DSR Measure focusing of offset image gathers by () scaling by h, () taking mean square: J DS [v;d] = hp n (m, h, ) m,h,n Presumption: if v kinematically consistent with data p focused at h = J DS [v;d] minimized over v. Theory: in ray-theoretic limit, presumption is correct (De Hoop & Stolk, ; De Hoop et al, 5) - need assume only absence of turning rays, consistent with DSR. [Note: analogous principle fails eg. for Kirchhoff common offset migration if multipathing occurs - imaging artifacts (Stolk & WWS 4).]

12 The Algorithm Because J DS is smooth and (perhaps) unimodal, can use rapidly convergent quasi- Newton algorithm. Limited memory variant of Broyden-Fletcher-Goldfarb-Shanno (Nocedal, 98 - available through Netlib); Velocity parametrization - bicubic splines, sigmoid representation to enforce bounds; Gradient computation - adjoint state method applied to DSR. [See abstract, references for details]

13 Examples

14 Example : Marmousi reflectivity, linear velocity Data (both examples) generated by time domain FD method for Born modeling. Source wavelet = Hz zero phase trapezoidal bandpass filter. Target velocity, used to generate data: linear, =.5 km/s at z =, = 4.5 km/s at z = 3km, represented on bicubic spline grid of 6 nodes in x ( x =.8 km) and 5 nodes in z ( z =.75 km). Initial velocity also linear, = km/s at z =, = 4.5 km/s at z = 3 km. Reflectivity = Marmousi velocity model (Versteeg and Grau, 99) minus m smoothing. Data geometry same as original. 3

15 Example : Marmousi reflectivity, linear velocity Position (km) 4 6 Position (km) 4 6 Left: Migrated image at initial v. Right: after LBFGS v updates. 4

16 Example : Marmousi reflectivity, linear velocity Position (km) 4 6 Position (km) Left: Input reflectivity ( true image). Right: image from DS velocity analysis. Good focusing and geometry in center. Some residual sag from initial velocity error remains on sides. Suggests that larger aperture more accurate v DS. 5

17 Example : Layered reflectivity, smoothed Marmousi velocity Position (km) 5 Position (km) 4 6 Modeling Inputs. Left: Smoothed Marmousi velocity (6 m smoothing width). Right: layered reflectivity. 6

18 Example : Layered reflectivity, smoothed Marmousi velocity Position (km) 5 Position (km) 4 6 Left: Linear velocity, initial guess for optimization. Right: image. 7

19 Example : Layered reflectivity, smoothed Marmousi velocity Position (km) 5 Position (km) 4 6 After LBFGS iterations. Left: velocity. Right: image. 8

20 Example : Layered reflectivity, smoothed Marmousi velocity Position (km) 5 Position (km) 4 6 Modeling Inputs. Left: Smoothed Marmousi velocity (6 m smoothing width). Right: layered reflectivity. 9

21 Example : Layered reflectivity, smoothed Marmousi velocity Offset (km) - Offset (km) - Offset (km) - Plot of image gather scaled by h and squared (sum of all such scaled, squared gathers = J DS ). Left: initial model. Center: LBFGS updates. Right: input ( true ) model. J DS for updated model actually smaller than for input model!

22 Example : Layered reflectivity, smoothed Marmousi velocity How can a wrong model generate better focus than a right model? Cause: coherent noise in image gathers. Possible causes of larger h signal in image gathers for true velocity: coherent noise in data [various remedies, but unlikely to be a factor here]; edge diffractions [remedy: taper on all axes]; mismigration of high angle events - consistent with appearance of gathers. [remedy: better propagator? Two-way RTM?].

23 Conclusions

24 Conclusions Have demonstrated a DSR-based automated VA prototype, extensible to 3D via common azimuth approximation. Cost of velocity analysis a few s of migrations. With present components, algorithm makes large model updates and greatly improves focusing of images. Available aperture affects accuracy (cf. tomography). Coherent noise in image volume, from data or imaging operator, can degrade accuracy. Action items for further research: assess effect of better (more expensive!) extrapolators in reducing operator-induced coherent noise; investigate effect and mitigation / modeling of coherent data noise; quantify aperture influence on velocity resolution. 3

25 Acknowledgements The authors are grateful to Total E&P USA for support of Alexandre Khoury s VIS research at Rice University, and for permission to present these results, to Henri Calandra for his encouragement and advice, and to Maarten de Hoop and Christiaan Stolk for many useful conversations. WWS thanks the sponsors of The Rice Inversion Project for their long-term support of related research on automation of velocity model building. 4