Kirchhoff Migration
Kirchhoff migration is a seismic imaging algorithm that reconstructs a true reflectivity image from recorded seismic data by summing (integrating) amplitudes along computed diffraction hyperbola curves for each output image point, based on the Kirchhoff integral solution to the scalar wave equation, with amplitude weighting functions accounting for geometric spreading, obliquity, and bandlimited derivative operators to produce a quantitatively accurate migrated image.
Key Takeaways
- Kirchhoff migration operates by computing the diffraction traveltime curve for every output image point using a velocity model, then summing input seismic traces along that curve; constructive interference of signal contributions at the correct image point and destructive interference elsewhere collapses diffracted energy back to its scattering source.
- Pre-stack Kirchhoff migration (PSTM and PSDM) operates on individual common-offset or common-angle gathers before stacking, preserving amplitude versus offset (AVO) attributes and enabling angle-dependent reflectivity analysis for fluid and lithology discrimination.
- The migration aperture controls which input traces are included in each summation; too narrow an aperture suppresses steeply dipping events and reduces structural resolution, while too wide an aperture includes noise and amplitude artifacts from far-offset head waves.
- Anti-aliasing filters applied along the summation curve prevent spatial aliasing artifacts from inadequately sampled input data, an important implementation detail that distinguishes robust from naive Kirchhoff implementations.
- Despite being computationally less accurate than reverse time migration (RTM) in complex overburden, Kirchhoff PSDM remains widely used due to its computational efficiency, flexibility with irregular acquisition geometries, and compatibility with target-oriented imaging workflows.
Fast Facts
Kirchhoff migration was first applied to seismic data processing in the 1970s, based on the mathematical framework established by Gustav Kirchhoff in the 19th century for wave diffraction. Pre-stack Kirchhoff depth migration (PSDM) became industry-standard in the 1990s for subsalt and complex structural imaging. A typical PSDM project for a 3D seismic survey of 1,000 square kilometers requires 10 to 100 teraflops of computational effort, compared with 100 to 1,000 teraflops for RTM on the same dataset. Kirchhoff migration is still estimated to account for more than 50 percent of commercial seismic depth migration production worldwide.
Tip: When designing a Kirchhoff PSDM workflow, invest heavily in the velocity model: the accuracy of the migration depends far more on the velocity field than on the choice between Kirchhoff and other migration algorithms in most geological settings. Kirchhoff migration with an accurate velocity model will outperform RTM with an inaccurate velocity model in imaging quality and AVO fidelity.
What Is Kirchhoff Migration
Seismic data recorded at the surface is not a direct image of the subsurface: reflections and diffractions from subsurface boundaries arrive at the recording array smeared in time and offset across many source-receiver pairs due to the geometry of wave propagation. Migration is the process of undoing this smearing to reconstruct the true spatial position and amplitude of each reflector. The term migration refers historically to the apparent movement of dipping reflectors from their unmigrated recording positions to their true geological positions in the migrated image.
Kirchhoff migration is named after its mathematical foundation: the Kirchhoff integral theorem, which expresses the wavefield at any point as a weighted surface integral of the wavefield and its normal derivative over a closed boundary. Applied to seismic imaging, this means that the reflectivity at any subsurface image point can be computed by summing the recorded wavefield at the surface over all source-receiver pairs, using traveltime weights computed from a velocity model to determine which surface data contribute to each image point.
How Kirchhoff Migration Works
The core operation of Kirchhoff migration for a single image point P consists of computing the two-way traveltime from each source to P and back to each receiver, looking up the amplitude on the corresponding seismic trace at that traveltime, applying amplitude and phase correction weights, and summing all such contributions. The traveltime computation uses ray tracing through the velocity model, producing a precomputed traveltime table (or "Green's function table") that maps each source-receiver-image-point triplet to a traveltime. The amplitude weights account for geometric spreading (amplitude decay with distance), obliquity (angle of the ray relative to the reflector normal), and the bandlimited derivative operator that converts displacement to reflectivity.
The summation is performed over all input traces within the migration aperture, a spatial domain around each image point defined by the maximum allowed ray angle from vertical. The aperture must be large enough to capture steep dips and provide full illumination of the image point, but excessively large apertures introduce noise and head-wave artifacts. Proper aperture design is one of the key quality control steps in Kirchhoff migration.
Post-stack Kirchhoff time migration operates on stacked seismic volumes and is fast but cannot correct for offset-dependent moveout errors or image through strong velocity contrasts. Pre-stack time migration (PSTM) operates on common-offset or common-midpoint gathers before stacking, preserving AVO information and partially correcting for lateral velocity variation. Pre-stack depth migration (PSDM) uses a full 3D velocity model to ray trace in depth, handling significant lateral velocity variation including the sub-salt imaging problem in the Gulf of Mexico or sub-basalt imaging in the North Atlantic. The transition from time to depth migration in the 1990s was the critical step that enabled commercial deepwater oil discoveries beneath thick allochthonous salt sheets.
Kirchhoff Migration Across International Jurisdictions
In Canada, Kirchhoff PSDM is standard for imaging complex foothills structures in the southern Alberta and British Columbia cordillera, where thrust belt geometries create severe ray bending and multivalued raypaths that require careful aperture design and ray tracing schemes beyond simple single-arrival Kirchhoff. WCSB deep basin operators use post-stack Kirchhoff time migration as a cost-effective processing choice for flat to gently dipping Cretaceous and Devonian targets where velocity complexity is limited. The AER does not prescribe specific migration algorithms; seismic processing decisions are the operator's technical responsibility, though final processed seismic data must be deposited with the Energy Resources Conservation Board on regulatory submission timelines.
In the United States, Kirchhoff PSDM transformed Gulf of Mexico deepwater exploration starting in the 1990s by enabling imaging beneath the extensive Sigsbee and other salt bodies that cover major hydrocarbon accumulations. BSEE does not regulate seismic processing methodology, but the competitive lease sale environment in the deepwater Gulf has driven major operators including ExxonMobil, Chevron, Shell, and BP to invest heavily in proprietary velocity model building and Kirchhoff or hybrid Kirchhoff-RTM migration workflows. In the Permian Basin, onshore Kirchhoff time migration of high-density 3D surveys guides horizontal well landing and fracture stimulation design in Wolfcamp, Spraberry, and Delaware Basin targets.
In Norway, Kirchhoff PSDM has been applied to most NCS 3D seismic surveys for structural imaging in the North Sea and Barents Sea. Sodir requires that reprocessed seismic data be submitted to the DISKOS national data center and remain accessible to other operators, creating a robust public seismic data archive. Equinor and CGG Norway have contributed research on multi-arrival Kirchhoff migration and wave-equation datuming methods to improve imaging under the complex overburden of the Horda Platform and the western Barents Sea shelf, where buried reefs and shallow gas create severe velocity anomalies.
In the Middle East, Saudi Aramco's exploration and reservoir imaging programs use Kirchhoff PSDM for structural interpretation beneath complex salt diapirs in the Red Sea and for high-resolution imaging of carbonate reservoir heterogeneity in the Eastern Province fields. The Arabian Shield basement is seismically simple compared to Gulf of Mexico salt, making PSTM or Kirchhoff PSDM with modest velocity models sufficient for most of Aramco's prolific onshore operations. For deep high-pressure targets and exploration in the Rub' al Khali basin, more advanced migration including anisotropic Kirchhoff PSDM accounting for VTI symmetry in the Jurassic shale overburden is deployed.
Synonyms and Related Terminology
Kirchhoff migration is also called diffraction summation migration, integral migration, or aperture migration in some contexts. Pre-stack variants are abbreviated PSTM (pre-stack time migration) and PSDM (pre-stack depth migration). Related concepts include seismic migration, reverse time migration (RTM), velocity model building, AVO (amplitude versus offset), common image gather, and wavefield extrapolation.
Frequently Asked Questions
Q: When should Kirchhoff migration be used instead of RTM?
A: Kirchhoff PSDM is preferred when computational budget is limited, when the acquisition geometry is irregular or sparse (Kirchhoff handles gaps well), when target-oriented imaging is needed (only a sub-volume is migrated), or when AVO angle gathers must be output efficiently. RTM is preferred when the velocity model contains strong lateral gradients such as salt flanks, when prismatic or refracted energy must be correctly migrated, or when the imaging target requires high-fidelity amplitude preservation through complex overburden.
Q: What causes Kirchhoff migration noise and how is it reduced?
A: Kirchhoff migration noise manifests as migration swings or operator aliasing artifacts, appearing as low-frequency smearing around image points. These artifacts arise from inadequate sampling of the summation curve (spatial aliasing), too-wide aperture including head wave energy, or insufficient anti-aliasing filtering. Reduction methods include aperture tapering at maximum dip angles, anti-aliasing pre-filtering of input data along the summation curve, and increasing the spatial sampling of the input data or of the traveltime tables through interpolation.
Why Kirchhoff Migration Matters
Kirchhoff migration is the workhorse of the seismic imaging industry because it produces quantitatively accurate subsurface images that can be directly interpreted for structure, stratigraphy, and reservoir properties. Without migration, seismic data cannot be reliably used for prospect mapping, well placement, or reservoir characterization; diffractions obscure structures, dipping reflectors appear at wrong positions, and fault imaging is unreliable. Kirchhoff PSDM in particular enabled the commercial development of deepwater plays globally by imaging through thick salt, basalt, and other complex overburden that was previously opaque to seismic methods. As the industry moves to machine learning-assisted seismic interpretation and digital subsurface workflows, migrated seismic volumes from robust algorithms like Kirchhoff PSDM provide the primary input data that all downstream interpretation products depend on.