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Refraction Tomography: A Practical Overview ofEmerging Technologies

Konstantin Osypov

WESTERNGECO, DENVER, COLORADO, USA

CSEG RECORDER | FEB 2001 | VOL. 26 NO. 02

ArticleBiosReferencesPrint

Near-surface velocity anomalies produce severe distortions in seismic images. If one knew the

detailed structure of these anomalies, the best way to tackle this problem would be to perform

wave-equation datuming or depth migration from the surface. However, 3-D prestack depth

imaging and datuming are computational challenges and highly sensitive to the near-surface

model. Therefore, statics application, assuming surface-consistent ray propagation through the

near surface, remains the most common approach to account for near-surface anomalies. For

estimating the short-period part of statics (so-called residual statics) analyzing the stacking

response of reflection data works well. However, estimating the long-period statics, using

reflection data alone, is more problematic. Therefore, solving for the near-surface velocity model

from refraction data, followed by calculating predominantly long-period statics, is customary in

seismic processing.

Refraction statics have long been implemented using delay-time methods. For example, the

generalized reciprocal method has been widely applied to 2-D data. Unfortunately, for 3-D

seismic, it is difficult to apply due to the lack of reciprocal data. However, the concept of delay

times is useful for 3-D refraction statics calculations by assuming first arrivals to be the onset of

head waves propagating along the refracting interfaces of locally flat layers on the scale of the

offset range (Figure 1). First-arrival picks are decomposed in a surface-consistent manner into

delay times and refracting-layer velocities. These are then converted to layer thicknesses

assuming a critical angle of incidence at the refracting layers. The model used in the delay-time

method is also used in head-wave tomography, as implemented, for example, in Generalized

Linear Inversion (Hampson and Russell, 1984). In this approach, instead of the two-step inversion

via delay times, traveltimes are inverted directly for layer thicknesses and velocities after ray

tracing of head waves.

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Figure 1: Model and rays for head waves.

Head-wave methods are in general robust because the relationship between the delay times and

the observed traveltimes is linear. However, in areas with complex geology and rough terrain the

layered model typically employed is often too simple to explain important data features.

Furthermore, since head-wave methods do not account for nonlinear moveout of first arrivals, it is

often necessary to limit the offset range. This limiting of the offset range contributes to a

fundamental velocity/depth ambiguity. To address this issue, it is common practice to specify the

weathering velocity prior to depth estimation (Hampson and Russell, 1984).

Recently, diving-wave, or turning-ray, tomography has become a popular alternative to

head-wave methods (Zhu et al., 1992; Bell et al., 1994; Osypov, 1998; Zhu et al., 2000). In this

approach, the medium is typically parameterized as a number of cells. Turning rays are then

traced through the model, and traveltime residuals are inverted for velocity perturbations in every

cell crossed by rays. Since this method adds more degrees of freedom to the model, it generally

fits the observed first-arrival moveouts better than headwave methods. Specifically, the model for

diving waves includes vertical velocity gradients (Figure 2). This accounts for the nonlinear

moveout of first arrivals and allows for the solution to incorporate a wider offset range. However,

the relationship between the model parameters and the traveltimes becomes nonlinear due to the

significant sensitivity of turning-ray paths to the velocity model. The nonlinearity is usually

handled by iterating the ray tracing and the model updating using a local linearization. This

makes the tomographic results sensitive to the initial model. Since the quality of the initial model

depends on the analyst’s expertise, this may cause a bias in the final solution. The above issues

are exacerbated when the data quality is poor. In other words, the results of diving-wave

tomography are usually more sensitive to pick errors than are the solutions of head-wave

methods.


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Figure 2: Model and rays for diving waves.

A desirable goal for diving-wave tomography is to reformulate it as a linear problem in order to

remove the initial-model dependency. Toward this goal, traveltime inversion in the τ-p domain is

very attractive, as it possesses an inherently linear formulation. Different forms of τ-p traveltime

inversion have long been used in earthquake seismology. However, the practical application to

3-D refraction tomography, until recently, was constrained by a 1-D assumption. Figure 3

illustrates a 1-D inversion producing a unique velocity/depth model from the observed traveltime

curve under the assumption of velocity increasing with depth. For the velocity model shown in

Figure 3d, the traveltime vs. offset curve is plotted in Figure 3a. Given the observed traveltimes

along this curve, there exist two particular approaches for solving the inverse problem to estimate

the implied velocity model. One option is to take a derivative of the traveltime curve to calculate

apparent slowness as a function of offset x (Figure 3c). The apparent slowness in the 1-D case

corresponds to the ray parameter, p. In the x-p domain, one can then apply the Herglotz-Wiechert

(H-W) formula (Aki and Richards, 1980) to derive the depths of penetration for diving waves

corresponding to the ray parameters for the different offsets. Since the p value corresponds to the

physical velocity at the turning point, this transformation provides an estimate of the desired

velocity/depth function in Figure 3d.


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Figure 3: Illustration of the 1-D traveltime inversion. (a) The traveltime curveas a function of offset x; (b) The representation of the traveltime curve in τ-pdomain; (c) The representation of the traveltime curve in x-p domain; (d)Velocity V vs. depth z corresponding to the traveltime curve in (a).Velocity/depth model in (d) can be estimated using (b) or (c) representations.

The other solution of interest for the inverse problem, is to calculate intercept times, τ, for each p

(Figure 3b). Then, one can apply another form of the H-W formula in the τ-p domain to estimate

the velocity/depth model in Figure 3d. A discrete form of this H-W transformation corresponds to

the ray geometry for head waves, yielding a multi-layer delay-time relationship. Numerically, the

HW transformation in the x-p domain is more accurate and stable for continuous velocity

functions. However, the transformation in the τ-p domain is better for treating velocity

discontinuities. This leads us to adopt a hybrid form of the H-W transformation to invert both head

and diving waves for a velocity model with both sharp velocity transitions and mild vertical

velocity gradients. This transformation may be extended to the case of velocity inversions, but

the solution becomes non-unique.

In order to extend this hybrid approach to 2-D and 3-D, one has to decompose the observed first-

arrival traveltimes into an equivalent τ-p representation. I introduced a form of refraction

tomography without ray tracing using a local, 1-D H-W transformation applied to the traveltime

curves obtained by the decomposition of first arrivals (Osypov, 1999). Later, I modified the

decomposition by exploiting the tomographic-like integrals coupled with a 3-D extension of the

hybrid H-W approach (Osypov, 2000). This founded the basis for a new τ-p refraction tomography

method.

I implement τ-p refraction tomography as a two-step process. First, I decompose the first-arrival

picks to a best-fit τ-p representation in 3-D using a linear inversion that does not require any

explicit ray tracing, cell parameterization, or initial model. The second step is a separate, local

inversion that involves the estimation of the 3-D velocity/depth model from the derived τ-p

representation. This model building process may incorporate a priori information such as upholes.

Figure 4: Tomographic velocity/depth model. The lines correspond to the


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regional interfaces between different geological layers interpreted from wellinformation.

Figures 4, 5, and 6 show the results of applying τ-p refraction tomography to a data example from

Tunisia (courtesy of Petro-Canada). The results demonstrate a good agreement with the

independent geologic and uphole information. Other examples comparing the accuracy and

robustness of different forms of refraction tomography may be seen in Osypov (1998, 1999, and

2000).

Figure 5: Stack after the application of statics corresponding to thetomographic model in Figure 4.

In conclusion, τ-p refraction tomography is an emerging technology that complements other

well-known methods for modeling the near surface and producing static corrections. This method

of traveltime inversion differs from conventional tomographic approaches in that it performs no

explicit ray tracing, although its formulation is implicitly based on ray theory. The approach

combines the robustness of delay-time methods, as it does not require an initial model, and the

flexibility of tomography, as it inverts both head and diving waves over the complete offset range.


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Figure 6: Comparison of vertical times through the tomographic model inFigure 4 with the uphole times.

About the Author(s)

Konstantin Osypov received his M.S. degree and Ph.D. in geophysics from St. Petersburg

University, Russia, in 1988 and 1992, respectively.

Prior to joining Western’s R&D in Denver in 1997 he worked at the University of Uppsala, Sweden,

and the Colorado School of Mines.

His main research interests lie in seismic tomography, inverse problems, statistical timeseries

analysis, and signal processing.

References

Aki, K., and Richards, P. G., 1980, Quantitative seismology: Theory and Methods, Freeman and Co.

Bell, M. L., Lara, R., and Gray, W. C., 1994, Application of turning-ray tomography to the offshore

Mississippi delta: 64th Ann. Internat. Mtg., Soc. Expl. Geophys., 1509-1512.

Hampson, D., and Russell, B., 1984, First-break interpretation using generalized linear inversion:

54th Ann. Internat. Mtg., Soc. Expl. Geophys., 532-534.

Osypov, K., 1998, A comparative study between 3-D diving-wave tomography and head-wave

refraction methods: 68th Ann. Internat. Mtg., Soc. Expl. Geophys., 1222-1225.

Osypov, K., 1999, Refraction tomography without ray tracing: 69th Ann. Internat. Mtg., Soc. Expl.

Geophys., 1283-1286.

Osypov, K., 2000, Robust refraction tomography: 70th Ann. Internat. Mtg., Soc. Expl. Geophys.,


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2032-2035.

Zhu, T., Cheadle, S., Petrella, A., and Gray, S., 2000, First-arrival tomography: method and

application: 70th Ann. Internat. Mtg., Soc. Expl. Geophys., 2028-2031.

Zhu, X., Sixta, D. P., and Angstman, B. G., 1992, Tomostatics: Turningray tomography + static

correction: The Leading Edge, 11, 15-23.

Appendices


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