Time to depth conversion and seismic velocity estimation using time migration velocities Journal: Manuscript ID: Manuscript Type: Date Submitted by the Author: Geophysics draft Letters n/a Complete List of Authors: Cameron, Maria; Univ. of California at Berkeley, Mathematics Fomel, Sergey; The University of Texas at Austin, Bureau of Economic Geology Sethian, James; Univ. of California at Berkeley, Mathematics Keywords: Area of Expertise: velocity analysis, migration, inversion, ray tracing, traveltime Geophysics Letters, Seismic Velocity/Statics
Page of GEOPHYSICS 0 0 0 0 0 0 Time to depth conversion and seismic velocity estimation using time migration velocities Maria Cameron, Sergey Fomel, and James Sethian Department of Mathematics, University of California, Berkeley, Berkeley, CA, 0 Bureau of Economic Geology, John A. and Katherine G. Jackson School of Geosciences The University of Texas at Austin University Station, Box X Austin, TX - (December 0, 00) Running head: Seismic velocity estimation ABSTRACT The objective of this work is to build an efficient algorithm (a) to estimate the seismic velocities from the time migration velocities, and (b) to convert the time migrated images to depth. We establish theoretical relations between the time migration velocities and the seismic velocities in D and D using paraxial ray tracing theory. The relation in D implies that the Dix velocities are the ratio of the seismic velocities and the geometrical spreading of the image rays. We formulate an inverse problem of finding seismic velocities from the Dix velocities and develop two numerical approaches for solving it. These approaches include Dijkstra-like Hamilton-Jacobi solvers for first arrival Eikonal equations and techniques for
Page of 0 0 0 0 0 0 data smoothing. We test these approaches on synthetic data examples and apply them to a field data example. We demonstrate that our algorithms give significantly more accurate estimate of the seismic velocity than the Dix inversion and that our estimate can be used as a good first guess in building velocity model for depth migration.
Page of GEOPHYSICS 0 0 0 0 0 0 INTRODUCTION Time-domain seismic imaging is a robust and efficient process routinely applied to seismic data (Yilmaz, 00). Rapid scanning and determination of time migration velocities can be accomplished, for example, by velocity continuation (Fomel, 00). Time migration is considered adequate for seismic imaging in areas with mild lateral velocity variations. However, even mild variations can cause structural distortions of time-migrated images and render them inadequate for accurate geological interpretation of subsurface stuctures. To remove structural errors inherent in time migration, it is necessary to convert timemigrated images into the depth domain either by migrating the original data with a prestack depth migration algorithm, by depth migrating post-stack data after time demigration (Kim et al., ), or by direct mapping from time to depth domains. Each of these options requires converting the time migration velocity to a velocity model in depth. The connection between time migration velocities and true depth velocity models is provided by the concept of image rays, introduced by Hubral (). Image rays are seismic rays arriving normal to the surface of the earth. Hubral s theory explains how to model time migration velocities given the depth velocity model. However, it does not provide a convenient form for developing an accurate inversion algorithm. Morever, tracing image rays is a numerically inconvenient procedure for achieving uniform coverage of the subsurface. This may explain why simplified image-ray tracing algorithms (Larner et al., ; Hatton et al., ) did not find widespread application in practice. The objective of the present work is to find an efficient method for building a velocity model from time migration velocities. We established new ray-theoretic connections between time migration velocities and seismic velocities in D and D. These results are based on the
Page of 0 0 0 0 0 0 image ray theory and the paraxial ray tracing theory (Popov and Pšenčik, ; Červený, 00; Popov, 00). In D, our result can be viewed as an extension of the Dix formula (Dix, ) to laterally inhomogeneous media. We show that the Dix velocities are seismic velocities divided by the geometrical spreading of the image rays. Hence, we use the Dix velocities instead of time migration velocities as a more convenient input. We develop two numerical approaches to find (a) seismic velocities from the Dix velocities, and (b) transition matrices from time domain coordinates to depth domain coordinates. We test the approaches on synthetic data examples and apply them to a field data example. TIME MIGRATION VELOCITIES Kirchhoff prestack time migration is commonly based on the following travel time approximation Yilmaz (00). Let s be a source, r be a receiver, and x be the reflection subsurface point. Then the total travel time from s to x and from x to r is approximated as T (s, x) + T (x, r) ˆT (x 0, t 0, s, r) () where x 0 and t 0 are effective parameters of the subsurface point x, and ˆT is an approximation, which usually takes the form: ˆT (x 0, t 0, s, r) = t 0 + x 0 s v m(x 0, t 0 ) + t 0 + x 0 r v m(x 0, t 0 ), () where x 0 and t 0 are the escape location and the travel time of the fastest image ray from the subsurface point x. Regarding this approximation, let us list four cases depending on the seismic velocity v and the dimension of the problem:. D and D, velocity v is constant. Equation is exact, and v m = v.
Page of GEOPHYSICS 0 0 0 0 0 0. D and D, velocity v depends only on the depth z. Equation is a consequence of the truncated Taylor expansion for the travel time around the surface point x 0. Velocities v m depend only on t 0 and is the root-mean-square velocity: v m (t 0 ) = t 0 t0 In this case the Dix formula (Dix, ) is exact. 0 v (z(t))dt. (). D, velocity is arbitrary. Equation is a consequence of the truncated Taylor expansion for the travel time around the surface point x 0. Velocities v m (x 0, t 0 ) are a certain kind of mean velocities, and we will establish what exactly they are below.. D, velocity is arbitrary. Equation is heuristic and is NOT a consequence of the truncated Taylor expansion. In order to write an analog of travel time approximation for D which is a consequence of the truncated Taylor expansion, we use the relation proven in Hubral and Krey (0): Γ = [v(x 0 )R(x 0, t 0 )], () where Γ is the matrix of the second derivatives of the travel times from a subsurface point x to the surface, R is the matrix of radii of curvature of the emerging wave front from the point source x, and v(x 0 ) is the velocity at the surface point x 0. For convenience, we prefer to deal with matrix K Γ, which, according to equation is K(x 0, t 0 ) v(x 0 )R(x 0, t 0 ). () The travel time approximation for D implied by the Taylor expansion is = ˆT (x 0, t 0, s, r) t 0 + t 0(x 0 s) T [K(x 0, t 0 )] (x 0 s) ()
Page of 0 0 0 0 0 0 + t 0 + t 0(x 0 r) T [K(x 0, t 0 )] (x 0 r). The entries of the matrix t 0 K(x 0, t 0 ) have dimension of squared velocity and can be chosen optimally in the process of time migration. In D, we will call the square roots of the entries of this matrix time migration velocities. SEISMIC VELOCITIES In this section, we will establish theoretical relations between time migration velocities and seismic velocities in D and D. Consider a set of image rays in D. Pick one of them and call it central. Let it arrive at a surface point x 0. Trace it backwards together with image rays arriving at a small neighborhood of the point x 0 for time t 0. Let the central ray reach a subsurface point x then. The behavior of the image rays close to the central one can be characterized by the matrices Q and P (Popov and Pšenčik, ; Červený, 00; Popov, 00). These matrices are defined as follows. Attach a coordinate system (t, q, q ) to the central ray, where t is the travel time along it, and q and q are the distances from it in particular two mutually orthogonal directions e, e lying in the plane normal to the central ray at the time t. Let p and p be the generalized momentums corresponding to the coordinates q and q. Define matrices Q and P by Q ij = q i α j, P ij = p i α j, i, j =,, () where (α, α ) are the coordinates of the initial surface point. The time evolution of the
Page of GEOPHYSICS 0 0 0 0 0 0 matrices Q and P is given by d Q dt = 0 v 0 I Q P, () v 0 V 0 P where v 0 it the velocity at the central ray at time t, V = ( v q i q j, and I is the )i,j=, identity matrix. The absolute value of det Q has a nice geometrical sense: it is the geometrical spreading of the image rays. Popov and Pšenčik (); Červený (00); Popov (00) showed that the matrix Γ relates to Q and P as Γ = PQ. Hence, K = QP. We have proven that t 0 K(x 0, t 0 ) = v (x(x 0, t 0 ))[Q(x 0, t 0 ) T Q(x 0, t 0 )], () where v(x(x 0, t 0 )) is the seismic velocity at the subsurface point x reached by an image ray traced backwards from a surface point x 0 for time t 0. In D, the matrices K, Q and P become scalars, and equation reduces to v(x(x 0,t 0 ),z(x 0,t 0 )) Q(x 0,t 0 ) = t 0 [t 0 v m(x 0, t 0 )] v Dix (x 0, t 0 ). K t 0 = (0) Thus, in D, the Dix velocity is the seismic velocity divided by the geometrical spreading of the image rays. The detailed derivation of equations and 0 is provided in (Cameron et al., 00). INVERSION METHODS FOR D Although we were able to prove finding seismic velocities from Dix velocities is ill-posed (Cameron et al., 00); nonetheless we were able to develop techniques which attempt the regularized reconstruction. One is based on the image ray tracing coupled with a fast
Page of 0 0 0 0 0 0 marching type time-to-depth conversion algorithm. The other is based on the level set method and includes the time-to-depth conversion algorithm in its time cycle. Ray Tracing Coupled with Time-to-Depth Conversion In our first algorithm, we begin by obtaining correct velocities v(x 0, t 0 ) in time-domain coordinates from Dix velocities v Dix (x 0, t 0 ) by solving the ray tracing system x t = v sin θ, z t = v cos θ, θ t = v q, () Q t = v P, P t = vqq v Q, where the subscript q denotes the differentiation in the direction orthogonal to the ray. The initial conditions are x(0) = x 0i, z(0) = 0, θ(0) = 0, Q(0) =, P (0) = 0. This is an ill-posed problem, and we smooth the found velocities and estimate the first and the second derivatives using the least squares polynomial approximation. This algorithm is very fast, however it fails if the image rays converge or diverge too strongly. Next, we propose a time-to-depth algorithm whose input is v(x 0, t 0 ) and the outputs are the seismic velocities v(x, z) and the transition matrices from time-domain to depthdomain coordinates x 0 (x, z) and t 0 (x, z). We solve the Eikonal equation with an unknown right-hand side coupled with the orthogonality relation t 0 = v(x 0 (x, z), t 0 (x, z)), t 0 x 0 = 0. () The orthogonality relation means that the image rays are orthogonal to the equitime curves. the buiding block for this algorithm is the Fast Marching Method (Sethian, ). Such time-to-depth conversion is very fast and produces the outputs directly on the depth-domain grid.
Page of GEOPHYSICS 0 0 0 0 0 0 Level Set Approach the second approach is based on the level set method embedding (Osher and Sethian, ). We define the front as the zero level set of a function φ(x, z) which we build and evolve in such a way that it always remains to be the signed distance function from the front. We also embed the quantities Q and P on the front into functions q(x, z) and p(x, z) so that q and p coincide with Q and P at the zero level set of φ(x, z). the input is Dix velocities v Dix (x 0, t 0 ) and the output is seismic velocities v(x, z). We solve the following system of PDE s: φ t0 + v(x, z) φ = 0, q t0 = v (x, z)p, () p t0 = vxxg x v xzg xg z+v zzg z v(x,z) q, where g x = φx φ, g z = φz φ. The algorithm has a complicated time loop:. estimate v(x, z) for current q using the time-to-depth algorithm;. detect the front and finding the velocity on the front (Sethian, );. build the extension velocity whose gradient is orthogonal to the signed distance function from the front (Sethian, );. estimate of the second derivatives of the extended velocity using the least squares polynomial approximation;. update φ, q and p using the extended velocity. This approach is significantly slower than ray tracing coupled with the time-to-depth conversion but it is able to handle successfully image rays intersection following the first arrival
Page 0 of 0 0 0 0 0 0 front. After the seismic velocity v(x, z) is found, one can use the Fast Marching Method (Sethian, ) to find the transition matrices x 0 (x, z) and t 0 (x, z). EXAMPLES Synthetic data example Figure (a) shows a synthetic velocity model. The model contains a high velocity anomaly that is asymmetric and decays exponentially. The corresponding Dix velocity mapped from time to depth is shown in Figure (b). There is a significant difference between both the value and the shape of the velocity anomaly recovered by the Dix method and the true anomaly. The difference is explained by taking into account geometrical spreading of image rays. Figure (c) shows the velocity recovered by our method and the corresponding family of image rays. Field data example Figure, taken from (Fomel, 00), shows a prestack time migrated image from the North Sea and the corresponding time migration velocity obtained by velocity continuation. The most prominent feature in the image is a salt body which causes significant lateral variations of velocity. Figure compares the Dix velocity converted to depth with the interval velocity model recovered by our method. As in the synthetic example, there is a significant difference between the two velocities caused by the geometrical spreading of image rays. Figure compares three images: post-stack depth-migration image using Dix velocities, post-stack depth-migration image using the velocity estimated by our method, and 0
Page of GEOPHYSICS 0 0 0 0 0 0 prestack time-migration image converted to depth with our algorithm. The evident structural improvements in Figure (b) in comparison with Figure (a) and a good structural agreement between Figures (b) and (c) is an indirect evidence of the algorithm success. An ultimate validation should come from prestack depth migration velocity analysis, which is significantly more expensive. CONCLUSIONS We have proved that the Dix velocity obtainable from the time migration velocity is the true velocity divided by the geometrical spreading of image rays. We have posed the corresponding inverse problem and suggested two algorithms for solving it. We tested these algorithms on a synthetic data example with laterally heterogeneous velocity and demonstrated that they produce significantly better results than simple Dix inversion followed by time-to-depth conversion. Moreover, the Dix velocity may qualitatively differ from the output velocity. We also tested our algorithm on a real data example and validated it by comparing prestack time migration image mapped to depth with post-stack depth migrated images.
Page of 0 0 0 0 0 0 REFERENCES Cameron, M., S. Fomel, and J. Sethian, 00, Seismic velocity estimation using time migration velocities: Inverse Problems, submitted for publication. Dix, C. H.,, Seismic velocities from surface measurements: Geophysics, 0,. Fomel, S., 00, Time-migration velocity analysis by velocity continuation: Geophysics,,. Hatton, L., L. K. Larner, and B. S. Gibson,, Migration of seismic data from inhomogeneous media: Geophysics,,. Hubral, P.,, Time migration - Some ray theoretical aspects: Geophysical Prospecting,,. Hubral, P. and T. Krey, 0, Interval velocities from seismic reflection time measurements: Soc. of Expl. Geoph. Kim, Y. C., W. B. Hurt, L. J. Maher, and P. J. Starich,, Hybrid migration: A costeffective -D depth-imaging technique: Geophysics,,. Larner, K. L., L. Hatton, B. S. Gibson, and I. S. Hsu,, Depth migration of imaged time sections: Geophysics,, 0. Osher, S. and J. Sethian,, Front propagating with curvature dependent speed: algorithms based on hamilton-jacobi formulations: Journal of computational physics,,. Popov, M. M., 00, Ray theory and Gaussian beam method for geophysicists: EDUFBA. Popov, M. M. and I. Pšenčik,, Computation of ray amplitudes in inhomogeneous media with curved interfaces: Studia Geoph. et Geod.,,. Sethian, J.,, A fast marching level set method for monotonically advancing fronts: Proceedings of the National Academy of Sciences,.
Page of GEOPHYSICS 0 0 0 0 0 0,, Level set methods and fast marching methods: Cambridge University Press. Červený, V., 00, Seismic ray theory: Cambridge Univ. Press. Yilmaz, O., 00, Seismic data analysis: Soc. of Expl. Geophys.
Page of 0 0 0 0 0 0 LIST OF FIGURES Synthetic test on interval velocity estimation. (a) Exact velocity model. (b) Dix velocity converted to depth. (c) Estimated velocity model and the corresponding image rays. The image ray spreading causes significant differences between Dix velocities and true velocities. Left: seismic image from North Sea obtained by prestack time migration using velocity continuation (Fomel, 00). Right: corresponding time migration velocity. Field data example of interval velocity estimation. (a) Dix velocity converted to depth. (b) Estimated velocity model and the corresponding image rays. The image ray spreading causes significant differences between Dix velocities and true velocities. Migrated images of the field data example. (a) Post-stack migration using Dix velocity. (b) Post-stack migration using estimated velocity. (c) Prestack time migration converted to depth with our algorithm.
Page of GEOPHYSICS 0 0 0 0 0 0 (a) (b) (c) Figure : Synthetic test on interval velocity estimation. (a) Exact velocity model. (b) Dix velocity converted to depth. (c) Estimated velocity model and the corresponding image rays. The image ray spreading causes significant differences between Dix velocities and true velocities. Cameron, Fomel, Sethian
Page of 0 0 0 0 0 0 Figure : Left: seismic image from North Sea obtained by prestack time migration using velocity continuation (Fomel, 00). Right: corresponding time migration velocity. Cameron, Fomel, Sethian
Page of GEOPHYSICS 0 0 0 0 0 0 (a) (b) Figure : Field data example of interval velocity estimation. (a) Dix velocity converted to depth. (b) Estimated velocity model and the corresponding image rays. The image ray spreading causes significant differences between Dix velocities and true velocities. Cameron, Fomel, Sethian
Page of 0 0 0 0 0 0 (a) (b) (c) Figure : Migrated images of the field data example. (a) Post-stack migration using Dix velocity. (b) Post-stack migration using estimated velocity. (c) Prestack time migration converted to depth with our algorithm. Cameron, Fomel, Sethian