2012 Honorary LecturerSponsored by Shell

Central & South America

Eduardo Filpo

PETROBRAS E&P/Exploration Rio de Janeiro, Brazil

Image ray time-to-depth conversion and model ray applications

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Eduardo Filpo

Although the use of PSDM images has been adopted as a standard procedure by the oil industry, imaging in time associated with time-to-depth conversion is still largely used. Time migration is a well-established technique to obtain useful time-domain images using stacking velocities available under routine seismic processing. Because of its robustness and simplicity, time-migrated sections provide a natural intermediate step before more ambitious and demanding depth migration. Although costing less in terms of computational and human effort, the time-to-depth converted image occasionally presents better quality than the depth-migrated one. The key for the right choice is understanding the validity and limitations of each algorithm used in the process.

When the use of time migration is valid, the connection between the time and depth domains is provided by image rays. The sample values within any given trace in the time-migrated section, are converted into depth by a corresponding image ray that starts vertically at the trace location and bends according to the Earth's velocity distribution.

However, time-to-depth conversion via Dix inversion and vertical stretching algorithms should be restricted because they have limited validity. They can produce good results only in the simplest case of a vertically inhomogeneous medium, where the image rays are vertical.

Time-migration algorithms can produce a good image when diffractions are well approximated by hyperbolas. In general, it happens when the Earth's velocity distribution is smooth, which is a favorable condition for using ray-tracing algorithms. Connection between points of a time-migrated section and its corresponding diffractor points in depth is given by the image ray starting at the corresponding midpoint location. Each seismic trace of the time-migrated section is associated with a specific image ray, while each isochron line is related to the wavefront perpendicular to all image rays in a specific traveltime. Image rays and wavefronts constitute an orthogonal coordinate system, referred as the image-ray coordinate system.

In general, when a time-imaging approach is applied, only an rms velocity field is available, and the image ray tracing is not possible due to the absence of a depth velocity model. The key is to use horizons to properly transform the rms velocity field into an interval velocity model in the time domain. Then, the interval velocity field in time can serve as input to an image wavefront-construction algorithm to trace the rays and the wavefronts which constitute the image-ray coordinate system. Once the image ray coordinate system is established, the time-to-depth conversion is reduced to a coordinate system transformation. The same approach can be applied to convert a PSTM volume, an interval velocity field, or a horizon from time to depth.  In this point of view, the time-to-depth conversion of seismic attributes can be performed as a simple coordinate transformation in which a vertical grid line of the Cartesian system (seismic trace) is mapped into an image ray and a horizontal line (time slice) into an image wavefront.

In the inverse problem (i.e., the depth-to-time conversion), vertical grid lines in depth are mapped into curves in the time domain. These curves are determined by a differential equation that is very similar to the eikonal equation. Because of this similarity, I suggest calling them model rays. Model rays are virtual trajectories in the time-migrated domain. There are two approaches to trace model rays, one based on the paraxial ray-tracing method and the other on wavefront construction by Huygens principle. The model ray tracing can be applied to solve several problems related to depth-to-time conversion, such as synthetic seismograms building, time conversion of PSDM images, well calibration, and construction of hybrid migration algorithms. 

Synthetic seismograms are widely used to match seismic data and well-log information. Lateral velocity variation around the borehole produces misfits of synthetic seismograms and trouble in wavelet estimation. Traditionally, the misfit problem is overcome by stretching and squeezing the synthetic trace, while wavelets are better estimated by using correlation windows instead of single traces. The main application of the model ray concept is the development of tools to improve the synthetic seismogram's matching by avoiding artificial deformation. The correct positioning of synthetic traces along model rays significantly reduces the need of stretching and squeezing, and it permits the use of smaller correlation windows for wavelet estimation.

Model rays have a strict relationship with image rays and play an important role in depth-to-time conversion problems. In summary, depth-to-time conversion can be implemented as a coordinate system transformation, where rectangular grids in depth become curvilinear in time (being vertical lines mapped into model rays) while horizontal lines are mapped into "same depth" curves which play the role of "model wavefronts" in time.