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The justification for smoothing well logs using the calculus is that we want to relate to surface seismic measurements by replacing the overburden with an equivalent homogeneous elastic medium. When using a smoothing method for ``relating'' well logs to surface seismic measurements, the length of the smoothing filter is the crucial parameter. One choice is to smooth over the smallest possible wave number determined by the surface method. This is the shortest wavelength we will see locally when a wave is propagating through a medium. We can average over that wavelength with the layer addition scheme, where average stress and strain components are calculated, which characterize the effective medium on a larger scale. It is important to notice that we are not smoothing traveltimes directly (like Dix more or less does), but rather assume that the measured traveltimes represent the vertical traveltimes only.
Further assuming isotropic layers (this is not necessary if we have more information) the P and S wave sonic log together with the density log specify an isotropic stiffness matrix. For each well log point in depth, these measurements are transformed into the group domain (, 1990). The important point is that all the smoothing is carried out in the group domain.
Equation 1 schematizes the steps from a conventional log to an ``in some sense'' equivalent log. The original log is converted to elastic stiffnesses , which in turn can be transformed in the group domain, whose elements are . An arbitrary operator can be applied in the group domain and the resulting group elements can be converted into quantities relating to a particular wave type and propagation direction. Here I just list a few: . Since the group transformation is a linear operation, we can apply any smoothing filter or other transform (Fourier transform, etc.) in this domain. All steps in equation 1 can be reversed as long as the operator itself is reversible.
It is not readily clear what the best smoothing operator is. I chose to use a boxcar smoothing operator. After smoothing the data in the group domain, the resulting effective medium will in general not be isotropic, but instead transversely isotropic. A useful elliptic approximation, as derived by Muir [(Muir,1991)], introduces in this way an effective NMO velocity. Figure 5: [ IMAGE ]
Figure 5 shows the ingredients for generating the isotropic stiffness log. From this stiffness log the group transformation is applied, resulting in a group log. Using this group log an equivalent medium can be calculated by partially integrating all the layers up to a certain depth point in the log, given by
This integration replaces the overburden at a specific depth point by a homogeneous equivalent medium. At the same time it preserves all stress and strain components across layer boundaries, honoring solid-solid boundary conditions exactly. For liquid layers different group elements have to be chosen, but the usage is identical. The elastic energy is conserved in this averaging process. See Nichols and Karrenbach [(Nichols and Karrenbach,1990)] for more details.
Figure 6: [ IMAGE ]
Figure 6 shows the well log derived Dix rms velocity and the horizontal and vertical velocities calculated using the partial integration. The Dix rms velocity is the solid curve, which is nearly identical with the horizontal velocity . The vertical propagation velocity is clearly distinct from these other two curves. It is not surprising that this partially-integrated medium is no longer isotropic. It is a well known fact [(Backus,1962)] that a sequence of thin layers produces anisotropic propagation effects for wavelengths much longer than the layering scale. Based on the assumption that the original material was thinly-layered isotropic, the partially-integrated medium is now transverse isotropic. I am only comparing P velocities since I started out with only a P sonic log and I had to assume a shear wave velocity. In Figure 6 I extracted only direct wave velocities, but using a paraxial approximation [(Dellinger \bgroup et al.\egroup ,1993)] it is possible to extract ``NMO'' velocities. For relating elastic properties to moveout velocities see the paraxial approximation around the vertical axis for TI media in the Appendix.
Figure 7: [ IMAGE ]
Figure 7 shows the Dix rms velocity and the ``wave averaged'' NMO velocity. The solid curve is the Dix rms velocity. The dashed curve is the horizontal NMO velocity calculated using partial integration. The dotted curve is the corresponding vertical NMO velocity . A marine surface seismic observation measures normal moveout velocity from a finite offset range, but leaves the vertical direct velocity ambiguous. As we can see in Figure 7, there is only a small difference at small depth, but growing disagreement with depth. The derived curves are always lower than the Dix rms velocity, as expected. The Dix rms velocity is a high frequency approximation satisfying Fermat's principle and thus provides us with the minimum travel time.
Figure 8: [ IMAGE ]
Figure 9: [ IMAGE ]
The crucial test however is to compare those curves to actual surface seismic data. Figure 8 shows a CMP gather right at the well site. Figure 9 shows a semblance velocity analysis of that same gather. Overlain are the two estimated velocity curves. The left most is the NMO velocity derived from the partially integrated stiffness log, while the right most curve is the Dix rms velocity. The velocity analysis is using only the raw data with a divergence correction, no other processing is applied. The multiples show up very strong in the semblance panel. Some primaries are visible and derived NMO velocity curves follow pretty close some primary semblance peaks at times less than 2.0 seconds. The Dix rms velocity seems off the semblance peaks to higher values, except for the water bottom reflection and close arrivals. Both curves miss a primary peak at around 1.5 seconds. This is a reflection from a dipping interface. Thus the velocity is shifted towards higher values than predicted using a horizontal layer assumption. Below the target around 2.0 seconds the peaks are shifted again towards higher values. Layers at that depth exhibit progressively steeper dips. The high contrast at the target zone gives rise to strong interbed multiples.
I summarize the velocity relationships for matching surface seismic velocity analysis with well log observations. Relationships between direct propagation velocity are:
The direct horizontal propagation velocity is essentially the same velocity component that is sampled by the P wave rms velocity. Thus their values are nearly identical. The horizontal direct propagation velocity is parallel to the layer interfaces, while the vertical direct propagation velocity perpendicular to the layering. The layering tends to lower the vertical with respect to the horizontal direct propagation velocity.
The rms velocity is derived from Fermat's principle (high frequency approximation) and thus has to fulfill the least travel time condition. The rms velocity is therefore higher than the vertical and horizontal NMO velocities. and are not necessarily least traveltime, they are derived from low frequency elastic constants and relate to the wave front curvature at small offsets, but additionally those velocities include effects caused by the micro structure of the medium, like multiple bounces and they are not least travel time measurements. Relationships between NMO velocities are:
Relationships between direct propagation and NMO velocities are:
Where the last relationship relates the two most interesting quantities for surface seismic, the direct vertical propagation velocity and the normal moveout velocity . In some special cases those two quantities can be made equal, but in general they are not (see the Appendix for such conditions). Figure 10: [ IMAGE ]
Figure 10 shows the results I obtained using Dix rms and the partial integration to relate well log measurements to surface seismic velocity analyses. Generally the predicted velocity is closer to the velocity picks. The largest difference in prediction by using rms velocities is 15 percent, while predictions are in the order of a few percent.
The previous examples all compare NMO and stacking velocities implying that the wave averages for that quantity with very large wavelength. It appears from the results that for velocity analysis the wave propagation effects tend to be better described using a low frequency averaging approach than a high frequency averaging approach.