Raypath

Key Takeaways

• Snell's law governs the bending of raypaths at every subsurface interface where seismic velocity changes: at an interface between a layer with velocity V1 above and V2 below, an incident ray arriving at angle theta1 (measured from the vertical normal to the interface) refracts to angle theta2 on the other side of the interface, with sin(theta1)/V1 = sin(theta2)/V2; when V2 is greater than V1 (the common case of velocity increasing with depth), theta2 is larger than theta1 and the refracted ray bends away from the vertical (toward horizontal); at the critical angle where theta2 equals 90 degrees (the ray travels horizontally along the interface), a head wave is generated that travels along the interface at velocity V2 and continuously radiates energy back to the surface as a refracted arrival used in refraction seismic surveys for shallow velocity determination; for incidence angles beyond the critical angle, total internal reflection occurs (no energy is transmitted into the slower medium), which is relevant for amplitude versus offset analysis because the reflection coefficient changes dramatically near the critical angle, and AVO attributes derived from pre-critical to post-critical angle reflections behave differently from those derived entirely within the pre-critical angle range; the critical angle for a typical sand-shale interface (V1 = 2,500 m/s shale over V2 = 3,000 m/s sand) is arcsin(2500/3000) = approximately 56 degrees, meaning that reflection data recorded at source-receiver offsets that produce incidence angles beyond 56 degrees at the target depth are in the post-critical regime where the standard AVO equations break down.
• Raypath geometry in a horizontally layered earth produces the hyperbolic travel time curve that is the mathematical basis of CMP stacking and velocity analysis: a reflection from a horizontal reflector at depth z, recorded at surface offset x from the source, follows a raypath that travels vertically downward from the source if the offset is zero (zero-offset ray), or at an angle for non-zero offsets; the total travel time T(x) for the reflected ray at offset x is given by T(x) = sqrt(T0^2 + x^2/Vrms^2), where T0 is the zero-offset two-way travel time and Vrms is the root-mean-square velocity from the surface to the reflector; this hyperbolic moveout relationship means that reflections on a CMP gather plot as hyperbolae in the offset-time domain, with the curvature of the hyperbola depending on the velocity (fast velocity = flat hyperbola, slow velocity = steep hyperbola); velocity analysis fits hyperbolae to the reflection moveout on CMP gathers to estimate Vrms as a function of two-way time, and the resulting velocity field is used to apply normal moveout correction (NMO) that flattens the hyperbola so that reflections at all offsets align in time, enabling coherent stacking; the hyperbolic moveout relationship is only exact for a single horizontal reflector in a constant-velocity medium, and in the real earth (with dipping layers, lateral velocity variations, and anisotropy) the moveout deviates from perfect hyperbolicity, requiring higher-order moveout corrections or full waveform inversion to handle correctly.
• Seismic migration uses raypath geometry to reposition reflectors from their apparent positions in the recorded data to their true subsurface positions, correcting for the fact that dipping reflectors are recorded at incorrect positions and that diffraction energy from subsurface edges and faults is spread across many traces in the unmigrated data: Kirchhoff migration (the most geometrically intuitive migration algorithm) propagates energy from each output image point backward along all raypaths to all source-receiver pairs at the surface, sums the contributions with appropriate time delays and amplitude weights (the Kirchhoff operator), and assigns the summed energy to the image point; the raypaths used in Kirchhoff migration are computed through a velocity model using Snell's law, so the migration accuracy depends directly on the accuracy of the velocity model: an incorrect velocity model produces incorrect raypaths, placing reflectors at the wrong depth and wrong horizontal position in the migrated image; ray-based migration methods (Kirchhoff and related algorithms) are computationally efficient and produce good results in areas with moderate velocity complexity, but fail in areas with strongly refracting velocity anomalies (beneath salt bodies, beneath shallow gas clouds, beneath highly overpressured zones) where ray-tracing through complex velocity models produces shadow zones, caustics, and multi-valued travel time fields that violate the single-valued raypath assumption underlying Kirchhoff migration; in these areas, wave-equation migration methods (reverse-time migration, RTM) that propagate full wavefields rather than raypaths are required.
• Amplitude versus offset analysis uses the angle of incidence along the raypath to relate observed reflection amplitude changes with offset to the elastic properties (P-wave velocity, S-wave velocity, and density) of the rocks on either side of the reflector: the Zoeppritz equations describe the exact relationship between angle of incidence and reflection and transmission coefficients for P-wave and S-wave energy at an interface, and their linear approximations (Shuey's two-term approximation, the Aki-Richards equation) are the standard tools used in exploration AVO analysis; the angle of incidence at the target reflector is determined from the source-receiver offset and the velocity model through ray-tracing (computing the raypath from source through the overburden to the reflector and measuring the angle at which the ray meets the reflector interface); in typical deep exploration targets (target depths of 3,000-5,000 meters), the maximum angle of incidence that can be achieved with practical source-receiver offsets (maximum offsets of 5,000-8,000 meters) is 30-45 degrees, which is sufficient to observe the near-to-far amplitude variation that distinguishes gas sands from brine sands in many geological settings; the accuracy of the AVO interpretation depends on the accuracy of the angle computation, which in turn depends on the accuracy of the velocity model used for ray-tracing, creating a chain of dependency from acquisition geometry through velocity estimation to AVO attribute reliability.
• Raypath complexity in the presence of salt bodies, shallow gas, and overpressured shales creates seismic imaging challenges that require sophisticated velocity model building and migration algorithms to overcome: in areas with subsalt targets (Gulf of Mexico, offshore Angola, offshore Brazil), compressional P-wave velocity through the salt body (approximately 4,480 m/s in halite) is much faster than in the surrounding sediments (typically 2,000-3,000 m/s), causing raypaths to bend dramatically toward the vertical as they enter the salt and back toward horizontal as they exit, creating shadow zones beneath salt overhangs where no reflected energy can be recorded from certain subsalt reflector positions; full azimuth wide-azimuth acquisition (WAZ or OBN acquisition with sources and receivers on all sides of the target) is used to fill in these shadow zones by providing raypaths from multiple directions that between them illuminate all subsalt positions; gas clouds in shallow sediments (generated by shallow gas migration from deeper sources) cause severe velocity anomalies that scatter seismic energy and create chaotic sections above and below the gas zone, requiring special acquisition designed to go around the gas cloud where possible, and careful velocity model building (often using full waveform inversion) to account for the gas cloud velocity anomaly in migration before the underlying reflectors can be imaged coherently.