Ray Tracing

Key Takeaways

• The two primary algorithmic approaches to seismic ray tracing are shooting (iterative ray bending to find the ray that connects a specific source-receiver pair) and bending (direct computation of the ray path geometry using analytical or semi-analytical equations for specific velocity model parameterizations): the shooting method starts a ray from the source at a trial angle, traces it forward through the velocity model using Snell's law at each interface, checks whether it arrives at the receiver location, adjusts the initial angle using a Newton-Raphson iteration, and repeats until the ray arrives at the receiver within a specified tolerance; the bending method (parameterized ray shooting, or the network shortest-path method based on Dijkstra's algorithm for graphs) is more computationally efficient for 3D models with many sources and receivers but may miss certain ray paths (particularly diving waves that turn gradually in velocity gradient layers) unless the method is extended to handle continuously varying velocity; pseudo-spectral methods (eikonal equation solvers including fast marching methods and level-set methods) compute the travel time field from a source to all points in the model simultaneously, providing the first-arrival time at every point in the model with a single computation pass, making them efficient for large-scale 3D models needed in seismic tomography and migration operator design.
• Seismic migration using ray-traced travel times (Kirchhoff migration) is the oldest and most widely used migration algorithm for producing structural seismic images from recorded data, using the ray-traced travel time from every source-receiver pair to every point in the image space to sum (stack) the seismic amplitudes at the appropriate time samples into a coherent image at each subsurface point: the Kirchhoff diffraction stack migration collapses the hyperbolic diffraction tails that appear on unmigrated seismic data (caused by point diffractors that scatter energy to many receiver locations) into their true subsurface positions by summing the trace amplitudes at the travel times predicted by ray tracing from the diffractor point to each source-receiver pair, and placing the summed amplitude at the diffractor point location in the migrated image; Kirchhoff migration handles steep dips and lateral velocity variations better than f-k (frequency-wavenumber) migration but is less accurate than wave-equation (finite-difference or spectral) migration methods for complex overburden with strong lateral velocity variations (salt bodies, gas chimneys) where the ray tracing approximation breaks down due to multipathing and wave front caustics (zones where rays cross and the single-arrival assumption of ray theory fails).
• Velocity model building for 3D seismic migration uses an iterative loop of ray tracing, residual moveout analysis, and tomographic updating to converge on a velocity model that correctly positions reflectors in depth: starting with an initial velocity model (typically derived from seismic stacking velocity analysis or from well check shots), ray tracing predicts the travel times of reflections from each depth point to each surface receiver; these predicted times are compared to the actual times observed in the common image point (CIP) gathers (the migration output sorted by reflection angle or offset), and the residual moveout (the difference between the flat CIP gather expected for a correct velocity model and the actual curved gather indicating velocity error) is converted back to velocity perturbations using ray-based tomography; the updated velocity model improves the flatness of the CIP gathers and the focusing of the migrated image, and the process is repeated until the velocity error is below a specified convergence criterion; this reflection tomography workflow, which critically depends on accurate ray tracing in the current velocity model at each iteration, is the standard velocity model building approach for pre-stack depth migration (PSDM) in complex geological settings including sub-salt imaging in the Gulf of Mexico and North Sea and sub-thrust imaging in fold-and-thrust belts.
• Refraction ray tracing for near-surface static correction models the propagation of head waves (refractions) along the base of the near-surface low-velocity layer (the weathering layer or LVL), where the refraction arrives at the surface before the direct wave for sufficiently distant source-receiver offsets; the refraction travel time as a function of offset follows a linear slope (with slope equal to 1/v_refractor, the velocity of the refractor that the head wave travels along) after a crossover distance that depends on the depth and velocity of the LVL; by fitting straight lines to the refraction arrival picks on multi-offset shot records, ray tracing of the refraction wave paths allows the thickness and velocity of the LVL to be estimated at each surface location, providing the near-surface corrections (static corrections) applied to the seismic traces to compensate for the travel time delay caused by the slow-velocity LVL before NMO correction and stacking; the accuracy of the refraction static model directly affects the stack quality and reflector continuity in the processed seismic data, because incorrectly corrected statics cause phase misalignment of reflections across the CMP gather that degrades the stack and produces mis-positioned reflectors on the migrated section.
• AVO modeling using ray tracing combines Zoeppritz equation amplitude calculations with the predicted ray parameter (angle of incidence) at each reflection point to compute the expected amplitude variation with offset (AVO) response for a specified elastic property model of the reservoir: at each interface in the layered Earth model, the Zoeppritz equations (or their approximate forms, the Shuey or Aki-Richards equations) relate the reflection coefficient to the angle of incidence and the contrast in P-wave velocity, S-wave velocity, and density across the interface; ray tracing determines the angle of incidence at each interface for each source-receiver offset, providing the input to the Zoeppritz calculation; the modeled AVO response is compared to the observed AVO from the seismic data to identify whether the observed amplitude variation is consistent with a gas-sand model (which typically shows increasing amplitude with offset for a soft-sand reflector), a brine-sand model (which shows different AVO behavior depending on rock physics), or a tuning effect (which can mimic AVO anomalies in thin beds); the ray-traced AVO model is a standard tool in prospect evaluation and DHI (direct hydrocarbon indicator) calibration, providing a quantitative test of whether the seismic amplitude anomaly is consistent with the elastic properties expected for a hydrocarbon-bearing reservoir.