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Z-99 / Converted-wave traveltime equations in layered anisotropic media: An overviewXiang-Yang Li1, Jianxin YuanLi2 and Richard Bale3
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1Edinburgh Anisotropy Project, British Geological Survey
2PGS Inc
3WesternGeco
Abstract
There are various types of converted-wave (C-wave) traveltime equations: for example, two-term or three-term Taylor series expansions and double-squared-root (DSR) equations. These different types of equations often have different kinds of parameterization and applicability. In this paper, we examine the accuracy and limitations of these equations and present improved equations under a unified system for multi-layered anisotropic media. These new forms of equations are accurate for far offsets up to twice the reflector depth, and can be used for velocity analysis, parameter estimation, and prestack imaging in anisotropic media. Field data examples are used to illustrate the applications.
Introduction
Converted-wave (C-wave) moveout is inherently non-hyperbolic, and has been intensively studied over recent years (e.g. Tsvankin and Thomsen, 1994; Cheret et al., 2000; Yuan and Li, 2001). One approach to this problem is the use of a higher-order Taylor series expansion. This approach is good for parameter estimation and makes it possible to process C-wave data independent of P-waves. Zhang (1996) proposed the use of a DSR equation, as an alternative approach, which treats the P- and S-wave legs in a C-wave raypath explicitly. Li and Yuan (1999) extended his approach to account for anisotropy. The DSR equation is more accurate for long-offset data, and can be used to perform C-wave prestack time migration (PSTM), but it depends on P-waves. Furthermore, the existing DSR equations are derived only for a single-layered medium. Here, we present an improved DSR equation for multi-layered media, and the equation is analytically derived from the anisotropic ray equation. We also re-formulate other equations using a unified notation. We compare the accuracy of all these equations with each other, and evaluate their merit for data processing.
Single VTI layer
Throughout the paper, Thomsen’s (1999) notation is used. Subscripts P, S, and C denote P-, S-, and C-waves respectively. Subscript i denotes interval quantities, subscript 2 denotes root-mean-squared (rms) quantities, and subscript 0 denotes vertical, or average quantities where appropriate. t stands for travel time, V for velocity, and g for velocity ratio. The type of anisotropy considered is transverse isotropy with a vertical symmetry axis (VTI). Anisotropic parameter (Alkhalifah, 1997), and two other parameters and are used. In a single VTI layer with Thomsen parameters and parameters and are defined as,
(1)
where 0, 2 and eff are vertical, stacking and effective P- and S-wave velocity ratios, respectively.
The travel time tC for a C-wave ray, converted at the bottom of the layer and emerging at offset x, can be derived as, in Taylor series form,
, (2)
where
(3)
is the anisotropic parameter defined in equation (1). The quartic coefficient A4 in equation (3) differs from Thomsen’s result (Thomsen, 1999) in that the denominator contains the velocity ratio 0, not (0+1).
Figure 1: Accuracy of the C-wave travetime equations. The residual moveout is caluclated as , and is calculated by ray tracing. The model contains an isotropic layer with VP=2500m/s, g0=2.5 and depth z=1000m. The four approximations are: squares – hyperbolic; crosses – Equation (9) in Thomsen (1999); triangles – single-layer DSR equation (4); circles – Equations (2) and (3).Similarly, the DSR equation for tC can be derived as,
, (4)
where
(5)
and xP and xS (xS=x-xP) are the horizontal distance of the corresponding P- and S-wave ray segments. Equation (4) also differs from Li and Yuan (1999) in that it contains anisotropic parameter , not and equation (4) is also more accurate. An analytical approximation for xP can also be derived from the ray equation (Li and Yuan 1999),
(6)
where
(7)
Many VTI layers
The Taylor series equation in multi-layered media has the same form as equation (2). The coefficients A4 and A5 are derived as (Yuan et al., 2001),
(8)
where,
, (9)
and eff and eff are effective anisotropic parameters, defined as, for the n-th layer,
(10)
Note that the relationship is no longer valid for multi-layered media. Equation (8) shows that the C-wave travel time in inhomogeneous, anisotropic media is controlled by four parameters These four parameters are: VC2, g0, geff and ceff. This forms the basis for parameter estimation.
The DSR equation is derived as,
Figure 2: The same as Figure 1, but for a 5-layer model, evaluated for the conversion from the bottom of the fifth layer. The squares, crosses, and triangles are the same as in Figure 1. The circles correspond to the full DSR equations (11) and (12), and asterisks correspond to the full Taylor expansion equations (2) and (8).(11)
where the expressions for tP, tS, xP and xS for the multi-layer case are the same as equations (5) and (6) for the single-layer case except coefficient c2. c2 for the multi-layer case is derived as,
(12)
heff and zeff are defined in equation (10).
Equations (11) and (12) show that the DSR equation in layered VTI media contains five parameters: VC2, g0, geff, heff and zeff . Note that heff and zeff are not independent parameters, and they are related to each other by equations (1) and (10). Equation (11) may not be very useful for parameter estimation, but it can be used for prestack migration.
Accuracy: Single-layer vs. multi-layer
In a single-layered medium, the DSR equations (4)-(7) are accurate up to offset-depth ratio (x/z) of 3.4 (triangle marks, Figure 1), and the corresponding Taylor series expansions (2) and (3) are only accurate up to x/z of 1.6 (circle marks, Figure 1). However, the original Thomsen’s (1999) approximation is only accurate to 1.0 (crosses, Figure 1). In multi-layered media, the DSR equations (11) and (12), and the Taylor series equations (2) and (8) have about similar accuracy up to x/z of 2.0 (circles and asterisks, Figure 2).
Parameter estimation using equations (2) and (8)
The DSR equations (11) and (12) contain five parameters, and cannot be used for parameter estimation directly. Here we investigate the use of the four-parameter equations (2) and (8) for parameter estimation.
The four parameters are: VC2, g0, geff and ceff. Numerical tests show that the travel time is less sensitive to the variations of the velocity ratios g0 and geff, and that even given VC2 and ceff, one can not resolve g0 and geff by semblance analysis. Thus g0 and geff have to be determined independently before semblance analysis. There are established work flows to determine g0 and geff. g0 is often determined by coarse correlation of the P- and C-wave stacked sections, and geff by CCP scanning, or joint inversion of the P- and C-wave stacking velocities. Once g0 and geff are determined, VC2 and ceff can be determined using a double-scanning procedure. This procedure is robust but time-consuming.
Data example
We use the Alba 4C data (MacLeod et al., 1999) to illustrate the application. Before anisotropic analysis, g0 and geff are determined first. Figure 3a shows the isotropic, non-hyperbolic NMO correction of a CCP gather from the Alba dataset. This is achieved using Equation (2), letting ceff=0. The events between 2.0 and 4.0 seconds are still not flattened at far-offsets by non-hyperbolic correction. A double-scanning procedure is then used to quantify VC2 and ceff. Figure 4 shows the double-scanning results for three selected events. ceff is well resolved with an average value of 0.25. Anisotropic NMO correction is then performed to the data with ceff=0.25, and the alignment of the events between 2.0 and 4.0 seconds are improved (Figure 3b).
For the event (tC0=3.9s) close to the Alba formation, the estimated parameters are: VC2=1300m/s, g0=2.9, geff=1.8 and c=0.25, giving rise to h=0.04, and z=0.13, the four Thomsen parameters as VP0=1983m/s, VS0=684m/s, e=0.08, and d=0.04. The values of and agree with the results from the borehole data in Cheret et al. (2000). This good agreement verifies the approach of anisotropic parameter estimation using reflection moveout. The estimated parameters (VC2, g0, geff and ceff) can then be used to build the anisotropic earth model (VP2, VS2, g0, heff and zeff),and the DSR equation can then be used for prestack migration.
(a) / (b)Figure 3: Comparison of different NMO corrections applied to a CCP gather of the inline component from the Alba 4C data. (a) Higher-order isotropic moveout correction using equation (2) and (3) with eff=0, and (b) same as (a) but with eff=0.25.
Discussion and conclusions
(a) / (b) / (c)Figure 4: Double scanning applied to the data in Figure 3 for different events at: (a) tC0=1574ms with g0=3.6 and geff=2.7, (b) tC0=2460ms with g0=3.2 and geff=2.3, and (c) tC0=3860ms with g0=2.9 and geff=1.8.
We have presented an unified system for traveltime equations in layered anisotropic media. These unified equations have captured the traveltime signatures accurately for mid-to-long offset data in terms of four measurable moveout attributes: VC2, g0, geff and ceff, and five medium parameters: VP2, VS2, g0, heff and zeff. The four moveout attributes can be determined from reflection moveout analysis and can then be used to build the anisotropic earth model for prestack time/depth migration.
Acknowledgements
We thank Chevron and the Alba partners for permission to show the Alba data. We thank Hengchang Dai for providing the Alba processing results. This work is funded by the Edinburgh Anisotropy Project (EAP) of the British Geological Survey, and the work is published with the approval of the director of BGS and EAP sponsors: Agip, BG, BP, ChevronTexaco, CNPC, ConocoPhillips, KerrMcGee, Landmark, Marathon, Norsk Hydro, PGS, Schlumberger, TotalFinaElf, TradePartners UK, Veritas DGC.
References
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