The application of the anisotropic energetic traveltime program on the SEG/EAGE salt body model:

Since 3-D transversely isotropic (TI) models are rare, we construct an example to test our 3-D traveltime calculator from the SEG/EAGE 3-D salt body model. The resulting TI model includes variable axis of symmetry direction that depends directly on the stratification direction. The strength and location of anisotropy in this model are derived from our experience with probable anisotropy behavior in field data. The steps taken here to build the anisotropic model can be used to derive anisotropic velocity models for purposes of anisotropic imaging in general.

The velocity model:

The SEG/EAGE velocity model was built under the sponsorship of the SEG and EAGE to have a 3-D velocity model suitable for testing 3-D prestack migration algorithms. Synthetic data were generated for the model, and migrations of these data have provided valuable insights into the 3-D imaging problem. Ultimately, the goal of prestack imaging of this model is to produce an accurate image of the subsurface, or in this case, an accurate image of the salt body and the underneath layers. The accuracy of the resulting image directly depends on the velocity model and the traveltime calculation. As a result, the salt body model can be used to test traveltime algorithms directly, which will give us some preliminary indications to the accuracy of the migration.

Figure 1 shows the SEG/EAGE salt body velocity model, with the shape of the salt body displayed clearly in the middle. Using this velocity model as an input into an accurate multi-arrival isotropic traveltime calculator should produce a perfect image of the salt body synthetic data. A reason for not obtaining a perfect image would be the presence of anisotropy ignored in the isotropic traveltime calculator. In this case, we will need to find the anisotropy parameters necessary to properly image the salt body. The velocity in Figure 1, thus, represents the NMO velocity obtained usually from stacking velocity analysis. Since, the vertical velocity in anisotropic media can not be estimated from surface seismic data, we will set it equal to the NMO velocity shown in Figure 1. The vertical velocity has little influence on imaging. It's main contribution is to position the reflectors accurately in depth. Such depth information can be extracted from well logs.

Fig. 1: A horizontal slice of the velocity field with the salt body displayed. The contour curves correspond to a vertical slice of the velocity field.

Next, we must estimate and build the Eta field, which the anisotropy parameter most responsible for imaging. For Eta=0 the medium is either isotropic or elliptically anisotropic, and isotropic imaging is usually enough to obtain a proper image.

The anisotropy parameter Eta:

The size of the anisotropy parameter Eta, the anisotropy parameter most responsible for the quality of imaging in TI media, often depends on the amount of shale present in the subsurface. Shale typically induce anisotropic behavior in seismic waves with size of anisotropy often depending on the depth of shale and the pressure it experiences. More than 90% of sediments in the Earth subsurface are shale, not all of which induce sizable anisotropy. Based on these observations, we build an Eta field for the salt body model in areas where we believe that shale layers may exist in practice. Figure 2 shows such an Eta field model with value of Eta increasing up to the salt body then dropping to zero.

Fig. 2: A vertical slice of the Eta field from the constructed 3-D Eta model. Also shown is the

salt body, which is, as typically the case, isotropic.

The vertical and NMO velocities and Eta define the anisotropy aspects of the TI model for P-waves (At least to the accuracy required in prestack imaging). If the symmetry axis is vertical no other parameters are needed to define the TI model. However, since the stratification in the Earth subsurface is not always horizontal, we can expect the symmetry axis to have some deviation from vertical especially around salt flanks.

The symmetry axis direction:

Two parameters are needed to define the symmetry axis direction in 3-D: an azimuth and an angle measured from vertical. The azimuth Phi measured from the x-axis ranges between -Pi/2 to Pi/2. It describes the direction of the vertical plane that contains the axis of symmetry. Within this vertical plane the angle Theta describes the symmetry axis direction. Theta also ranges from -Pi/2 to Pi/2 and it is measured from the vertical z-axis. Both angles, when provided, describe a unique direction in 3-D.

Since most anisotropies are due to the stratification and sedimentation of the subsurface, the symmetry axis is expected to have a direction orthogonal to the layering. Using the original SEG/EAGE salt body model, we can estimate probable symmetry axis directions by evaluating the direction of the gradient of the velocity. Figures 3 and 4 show the Phi and Theta angle fields which directly describe the direction of the gradient of the velocity field, with all angles given in radians. Figure 4 shows that departure of the symmetry axis from vertical ranges between -0.27 and 0.18 radians (between -16 and 10 degrees). This is a small but practical range. However, the traveltime algorithm could handle any range. From Figure 3, we note that the sediments on opposite sides of the salt body (left and right) are dipping in opposite directions. Such opposite direction dips can be observed on the contour plot in Figure 1 as well.

Fig. 3: A horizontal slice the the Phi angle field with salt body displayed in the middle.

Fig. 4: A horizontal slice the the Theta angle field with salt body displayed in the middle.

Traveltimes:

Inserting the above parameter fields in the anisotropic version of the energetic traveltime calculator provides us with a traveltime field that corresponds to a specified source position. The traveltime calculator can provide us with the most energetic traveltimes, the shortest path traveltimes, or all arrivals if needed. Figure 5 shows the shortest path arrivals for the anisotropic model given by the light blue contour curves. The purple contour curves correspond to the traveltime field extracted from the isotropic version of the code using only the velocity field shown in Figure 1. The difference between the isotropic and anisotropic traveltimes increases with ray dip angle. Since the same vertical velocity were used for both the isotropic and anisotropic models the traveltime contours practically coincide for rays traveling vertically. In practical cases the vertical velocity can be different from the stacking velocity. causing traveltime differences in the vertical direction as well. Despite the relatively thin anisotropic layer used here, the influence of anisotropy extends to waves traveling beneath the salt body with difference exceeding two percent in some cases.

Fig. 5: Vertical slices of traveltime contours in blue for the anisotropic model and in purple for the isotropic model. The isotropic model has the same velocity as the anisotropic one but a zero eta.

Figure 6 shows horizontal slices of the two traveltime fields (isotropic and anisotropic) given in contours at to different depths. The difference between the two traveltime fields decreases with depth, especially beneath the salt body. The rays are terminated when the ray dip exceeds 135 degree from vertical. The traveltime for such rays are, thus, not used and the traveltimes at the grid points corresponding to these areas are set to zero. This causes the contour gaps shown in Figures 6 and 7.

Fig. 6: Horizontal slices of traveltime contours in green for the anisotropic model and in white for the isotropic model superimposed over a horizontal slice of the velocity field. Left: a top view of the horizontal slice corresponding to a shallow depth. Right: a side view of the horizontal slice corresponding to a large depth.

Figure 7 shows a horizontal slice of the traveltime field which clearly displays the areas of successful traveltime calculations. Regions with energetic rays traveling beyond 135 degrees are set to zero. The domain of successful traveltime measurements is large and it is beyond what is typically needed for Prestack migration. Often rays with propagation dips of 80 degrees are enough for prestack migration application. The excessive range of ray dips used here is just for display reasons in this example.

Fig. 7: A horizontal slice of the traveltime field (in seconds) corresponding to the anisotropic model. The regions given by blue are zero times and they correspond rays having propagation angles (measured from vertical) beyond 135 degrees with the vertical.

Velocity analysis:

One of the most challenging problems in seismic processing is estimating the anisotropic parameters necessary for prestack migration. Though recent developments have simplified this problem in media with at most vertical velocity variations, velocity estimation in anisotropic complex media is still a puzzle to most of us. We have had the capability to obtain traveltimes for a given trial velocity model in anisotropic media. We can also obtain traveltimes or moveout residuals based on the measured traveltimes. The problem in anisotropic media is how to update the velocity model (or models) given these residuals.

Fig. 8: A vertical slice of the Eta field from the 3-D Eta model.

Fig. 9: Vertical slices of traveltime contours in blue for the anisotropic model and in purple for the isotropic model. The isotropic model has the same velocity as the anisotropic one but a zero eta. The salt body is also displayed.

Here we propose a priori information based approach, taking advantage of the flexibility achieved using the anisotropic traveltime calculator. Since shale layers induce anisotropy and the location of massive shale formations are often known in advance (from geological investigations), we can construct anisotropic layers in the model based on these formations by simply setting Eta to values larger than zero. Figure 2 shows an example of such a layer which represents areas of proposed shale formations. However, setting a value for Eta in that layer remains a mystery. Our only option is to try different trial Eta values and evaluate the resulting images. Figure 8 shows a new Eta model, similar in distribution to that in Figure 2, but different in absolute value. Specifically, the new Eta values are half of those in Figure 2. Figure 9 shows the corresponding traveltimes and its departure from the isotropic traveltimes. Since we are using lower Eta values in the new model the traveltime difference between the anisotropic model and the isotropic one is smaller in Figure 9 than that is Figure 3. The new anisotropic traveltime field shown in Figure 9 will result in a different image than that in Figure 3 and different than that of the isotropic. More trial models can be constructed this way and the one that produces the best image should be kept. Well logs provide us with vertical velocity information, which can be used in the anisotropic traveltime calculator to correctly position reflectors in depth while leaving the obtained best image intact. Such a feature does not exist in isotropic traveltime calculators.

Such flexibility is one of the main features of the anisotropic energetic traveltime calculator. The good news is that such flexibility is based on an actual physical phenomena of seismic waves referred to as anisotropy. A good image quality can be obtained by fudging the isotropic operator to fit the data. The problem is that the fudged parameters do not have any physical meaning in terms of the subsurface content..

Advantages:

The anisotropic traveltime calculator provides us with flexibility that lacks isotropic traveltime calculators. In the isotropic case, if the velocity field does not produce an adequate image our only option is to modify the velocity even if such modifications go against our common sense of how the subsurface behaves. Such modifications might also go against measured velocity information from well logs. Anisotropic traveltime calculators, on the other hand, can absorb such variations in the velocity field under a physical umbrella that has direct interpretation with respect to the Earth subsurface.

This traveltime calculator handles TI models with variable symmetry axis in 3-D. The only requirement is that the anisotropy parameters be smoothed, a common need of ray theoretical based traveltime computation. The ray equations used in the traveltime calculator are the simplest and most efficient out there. It is based on newly developed simplifications to the TI ray equations using the acoustic assumption