Alexey Stovas

Traveltime approximations for a layered transversely isotropic medium

We consider multiple transmitted, reflected, and converted qP-qSV-waves or multiple transmitted and reflected SH-waves in a horizontally layered medium that is transversely isotropic with a vertical symmetry axis (VTI). Traveltime and offset (horizontal distance) between a source and receiver, not necessarily in the same layer, are expressed as functions of horizontal slowness. These functions are given in terms of a Taylor series in slowness in exactly the same form as for a layered isotropic medium. The coefficients depend on the parameters of the anisotropic layers through which the wave has passed, and there is no weak anisotropy assumption. Using classical formulas, the traveltime or traveltime squared can then be expressed as a Taylor series in even powers of offset. These Taylor series give rise to a shifted hyperbola traveltime approximation and a new continued-fraction approximation, described by four parameters that match the Taylor series up to the sixth power in offset. Further approximations give several simplified continued-fraction approximations, all of which depend on three parameters: zero-offset traveltime, NMO velocity, and a heterogeneity coefficient. The approximations break down when there is a cusp in the group velocity for the qSV-wave. Numerical studies indicate that approximations of traveltime squared are generally better than those for traveltime. A new continued-fraction approximation that depends on three parameters is more accurate than the commonly used continued-fraction approximation and the shifted hyperbola.

Generalized nonhyperbolic moveout approximation

Reflection moveout approximations are commonly used for velocity analysis, stacking, and time migration. A novel functional form for approximating the moveout of reflection traveltimes at large offsets is introduced. In comparison with the classic hyperbolic approximation, which uses only two parameters (zero-offset time and moveout velocity), this form involves five parameters that can be determined, in a known medium, from zero-offset computations and from tracing one nonzero-offset ray. It is called a generalized approximation because it reduces to some known three-parameter forms with a particular choice of coefficients. By testing the accuracy of the proposed approximation with analytical and numerical examples, the new approximation is shown to bring an improvement in accuracy of several orders of magnitude compared to known analytical approximations, which makes it as good as exact for many practical purposes.

Reflection and transmission responses of a layered isotropic viscoelastic medium

Transmission effects in the overburden are important for amplitude versus offset (AVO) studies and for true‐amplitude imaging of seismic data. Thin layers produce transmission effects which depend on frequency and slowness. We consider an inhomogeneous viscoelastic layered isotropic medium where the parameters depend on depth only. This takes into account both the effects of intrinsic attenuation and the effects of the layering (including changes in attenuation). The seismic wavefield is decomposed into up‐ and downgoing waves scaled with respect to the vertical energy flux. This gives important symmetry relations for the reflection and transmission responses. For a stack of homogeneous layers, the exact reflection response can be computed in a numerically stable way by a simple layer‐recursive algorithm.The reflection and transmission coefficients at a plane interface are functions of the complex medium parameters (depending on frequency) and the real horizontal slowness parameter. Approximations for wea...

Estimation of layer thickness and velocity changes using 4D prestack seismic data

In some hydrocarbon reservoirs, severe compaction of the reservoir rocks is observed. This compaction is caused by production and is often associated with stretching and arching of the overburden rocks. Time-lapse seismic data can be used to monitor these processes. Since compaction and stretching cause changes in layer thickness as well as seismic velocities, it is crucial to develop methods to distinguish between the two effects. We introduce a new method based on detailed analysis of time-lapse prestack seismic data. The equations are derived assuming that the entire model consists of only one single layer with no vertical velocity variations. The method incorporates lateral variations in (relative) velocity changes by utilizing zero-offset and offset-dependent time shifts. To test the method, we design a 2D synthetic model that undergoes severe reservoir compaction as well as stretching of the overburden rocks. Finally, we utilize the method to analyze a real 2D prestack time-lapse seismic line from t...

Generalized moveout approximation for qP- and qSV-waves in a homogeneous transversely isotropic medium

The moveout approximations can be used in kinematic modeling, velocity analysis, and time migration. The generalized moveout approximation involves five approximation parameters and has several known approximations as special cases. A method is demonstrated for determining parameters of the generalized nonhyperbolic moveout approximation for qP- and qSV-waves in a homogeneous transversely isotropic medium with vertical symmetry axis (VTI medium). The additional parameters for the generalized approximation are computed from the hyperbolic asymptote at infinite offset. Comparison with a few well-known moveout approximations for higher-order terms in the Taylor series and asymptotic behavior shows that the generalized moveout approximation is superior to other nonhyperbolic approximations. A few numerical examples for qP- and qSV-waves in a VTI medium also indicate that the generalized approximation performs the best.

On the orientation and absolute phase of marine CSEM receivers

Receiver orientation can be recovered from electric and/or magnetic data if it is not directly measured. A receiver dropped on the seabed will end up with an arbitrary orientation, which means that the recorded electric and magnetic x- and y-components will point in arbitrary directions. We demonstrate how both electric and magnetic data can be used to rotate the field data to a coordinate system where the x-direction points in the inline or towline direction or 180° with respect to this direction. The amplitudes of electric and magnetic marine CSEM data are highly offset dependent so we introduce a median filtering approach to handle this problem. An inspection of the electric and/or magnetic phase after normalization with the source-current phase can resolve the remaining problem of the 180 degree spatial rotation. The result is electric and magnetic data where the x-component points in the positive towline direction. We analyze the case of lost temporal synchronization between receivers and the transmi...

Vertical propagation of low-frequency waves in finely layered media

Multiple scattering in finely layered sediments is important for interpreting stratigraphic data, matching well-log data with seismic data, and seismic modeling. Two methods have been used to treat this problem in seismic applications: the O’Doherty-Anstey approximation and Backus averaging. The O’Doherty-Anstey approximation describes the stratigraphic-filtering effects, while Backus averaging defines the elastic properties for an effective medium from the stack of the layers. It is very important to know when the layered medium can be considered as an effective medium. In this paper, we only investigate vertical propagation. Therefore, no anisotropy effect is taken into consideration. Using the matrix-propagator method, we derive equations for transmission and reflection responses from the stack of horizontal layers. From the transmission response, we compute the phase velocity and compare the zero-frequency limit with the effective-medium velocity from Backus averaging. We also investigate how the transition from timeaverage medium to effective medium depends on contrast; i.e., strength of the reflection-coefficient series. Using numerical examples, we show that a transition zone exists between the effective medium low-frequency limit and the time-average medium high-frequency limit, and that the width of this zone depends on the strength of the reflectioncoefficient series.

AVO attribute inversion for finely layered reservoirs

Assuming that a turbidite reservoir can be approximated by a stack of thin shale-sand layers, we use standard amplitude variaiton with offset (AVO) attributes to estimate net-to-gross (N/G) and oil saturation. Necessary input is Gassmann rock-physics properties for sand and shale, as well as the fluid properties for hydrocarbons. Required seismic input is AVO intercept and gradient. The method is based upon thin-layer reflectivity modeling. It is shown that random variability in thickness and seismic properties of the thin sand and shale layers does not change significantly the AVO attributes at the top and base of the turbidite-reservoir sequence. The method is tested on seismic data from offshore Brazil. The results show reasonable agreement between estimated and observed N/G and oil saturation. The methodology can be developed further for estimating changes in pay thickness from time-lapse seismic data.

Accuracy comparison of nonhyperbolic moveout approximations for qP-waves in VTI media

There are many non-hyperbolic moveout approximations for VTI media with various accuracy claims. We perform an analytical and numerical comparison of the 11 moveout approximations developed for a homogeneous transversely isotropic medium with vertical symmetry axis (VTI). The comparison is based on evaluating their long-offset asymptotes and higher-order Taylor series coefficients computed for a VTI medium. To make the different approximations comparable we select approximation coefficients to preserve the fourth-order coefficient in Taylor series for traveltime squared. To make the comparison simpler, we define the traveltime as a function of normalized offset and one parameter which is the anelliptical parameter. We select the six best approximations and perform comparative analysis by plotting the asymptotic traveltime parameters and high-order Taylor coefficients versus anelliptical parameter. We also compute the relative error in traveltime for selected approximations for homogeneous and horizontally layered VTI models.

Kinematically equivalent velocity distributions

For a layered medium, the seismic velocity model can be vertically heterogeneous within the layers. The traveltime parameters estimated from each reflection must be converted into layer traveltime parameters by using the layer-stripping method. The layer traveltime parameters must be inverted into layer velocity model parameters. Interpretation or inversion of layer traveltime parameters depends on the chosen velocity model within the layer. Different or kinematically equivalent velocity distributions can result in the same traveltime parameters. The inversion problem for traveltime parameters is strongly nonunique even if they are estimated accurately. To evaluate the accuracy of a velocity model, one can choose the phase for the two-way propagator. The discrepancy in this phase factor between the kinematically equivalent velocity models depends on the number of traveltime parameters estimated and increases with spatial frequency. By estimating two traveltime parameters, we approximately preserve the average velocity, regardless of the complexity of the vertically heterogeneous model. By estimating three traveltime parameters, we approximately preserve the average velocity gradient.