Kirchhoff Migration
Concept:
- Kirchhoff migration is a seismic imaging technique based on the Kirchhoff integral theorem. It's widely used because it can handle complex geology and irregular acquisition geometries. Kirchhoff migration can be applied in post-stack or pre-stack modes, depending on when the migration is performed relative to the stacking process.
- Based on Huygens’ principle every point on a wavefront is a source of secondary wavelets.
Post-Stack Kirchhoff Migration
When it's used:
After normal moveout (NMO) correction and stacking of common midpoint (CMP) gathers.
Input:
- Stacked seismic section
- Stacking velocity model (smooth velocity)
Output:
- Migrated seismic image with reflectors repositioned to their true subsurface locations.
Pre-Stack Kirchhoff Migration
When it's used:
Before stacking, on individual traces or CMP gathers, typically in pre-stack time migration (PSTM) or pre-stack depth migration (PSDM).
Input:
- Unstacked seismic data (e.g., shot gathers or CMP gathers)
- Detailed velocity model (can be time or depth dependent)
Output:
- High-resolution seismic image that preserves angle-dependent amplitude and geometry
Step-by-Step: Kirchhoff Time Migration
1. Input Data Preparation
- Seismic stacked section: A 2D array with time (vertical) and trace position (horizontal).
- Velocity model: A 2D array of stacking velocities that match the seismic data.
2. Select Output Grid
- Define the output image grid, usually the same as the input section (time vs. trace position).
3. Loop Over Output Points
For each point (x₀, t₀) in the output migrated section:
4. Compute Travel Time from Source to Scatter Point
- Since it's post-stack data, assume zero-offset (source and receiver are at the same point).
- Use the stacking velocity to calculate the total travel time from the image point to each data trace
5. Interpolate Amplitude
- From the input seismic section, interpolate the amplitude at time T on trace xxx.
6. Apply Weighting
- Apply amplitude corrections like geometrical spreading, anti-alias filtering, and obliquity factor (optional in time migration).
7. Stack Contributions
- Sum (or integrate) the weighted amplitudes over all traces for each output point
8. Output Migrated Section
- After all grid points are processed, the result is a migrated image, with reflectors moved to their correct spatial positions.
Strengths:
1. Computational Efficiency
- Fastest migration method for most cases (O(N²) complexity vs. O(N³) for wave-equation methods). because uses ray tracing and summation along diffraction hyperbolas rather than full wavefield modeling.
2. Flexibility in Handling Irregular Geometry
- Easily accommodates uneven trace spacing, missing data, or complex acquisition geometries.
3. Amplitude Preservation (When Properly Implemented)
- Can preserve relative amplitudes if weights (e.g., obliquity, spherical divergence) are correctly applied.
- Use Case: AVO (Amplitude vs. Offset) analysis.
4. Target-Oriented Imaging
- Can image specific subsurface targets without migrating entire datasets because summation is performed only for selected output points.
5. Robustness in High-Contrast Media
- Less prone to artifacts from sharp velocity contrasts compared to some wave-equation methods.
Limitations:
1. High-Frequency Approximation
- Assumes infinite-frequency rays, ignoring finite-bandwidth effects. So, poor handling of diffraction tails and subtle structural details.
2. Multi-Pathing Issues
- Cannot handle multiple ray paths (e.g., in complex velocity fields). So, mispositioned events in areas with strong velocity variations.
3. Amplitude Distortions
- Amplitude inaccuracies due to approximate weighting functions. Example: Underestimates amplitudes at steep dips.
4. Dip Limitations
- Struggles with steep dips (>60°) due to asymptotic approximations. Because of ray theory breaks down near critical angles.
5. Noise Sensitivity
- Summation along hyperbolas amplifies coherent noise (e.g., multiples).
- Requires careful pre-processing (e.g., demultiple).
Result of Kirchhoff Time Migration with AI
Result of Kirchhoff Depth Migration with AI
Comparison of Kirchhoff Time Migration result of AI with Industrial software
F-K (Stolt) Migration
Concept:
- A fast frequency-wavenumber (f-k) domain method ( also called Stolt migration ), often applied in time migration.
Strengths:
- 1. Computational Efficiency
Operates in the Fourier domain (F-K), leveraging the Fast Fourier Transform (FFT) for speed. - O(N log N) complexity vs. O(N²) for time-space methods.
2. Exact Solution for Constant Velocity
- Perfectly collapses diffractions when velocity is homogeneous.
- No numerical dispersion artifacts.
3. Parallelization-Friendly
- Each frequency-wavenumber component is processed independently.
4. No Dip Limitations
- Correctly handles steep dips (unlike some finite-difference methods).
5- Amplitude Preservation
- Maintains relative amplitudes better than Kirchhoff migration in simple models.
Limitations:
1. Velocity Model Restrictions
- Primary weakness: Assumes vertically varying velocity (1D).
- Fails with strong lateral velocity variations (e.g., salt bodies, faults).
2. Stretch Artifacts
- Time-to-depth conversion can distort wavelet shapes.
3. Edge Effects
- Circular convolution artifacts due to FFT periodicity (requires padding).
4. Limited Imaging Flexibility
- Hard to incorporate advanced features like:
- Anisotropy
- Attenuation (Q)
- Multi-pathing
5. Pre-Processing Sensitivity
- Requires careful data regularization and interpolation for irregular geometries.
Result of Frequency-Wavenumber(F-K) Migration with AI
Comparison of Frequency-Wavenumber(F-K) Migration result of AI with Industrial software
F-X (Omega-X) Migration
Concept:
- Works in the frequency-space (F-X) domain (Fourier over time, spatial in x/z).
- Each frequency component is migrated independently.
Strengths:
1. High Accuracy for Complex Structures
- Better at handling steep dips and complex geology than Kirchhoff migration.
- Resolves overlapping reflections more accurately due to wave-equation-based propagation.
2. Handles Lateral Velocity Variations Well
- Unlike Stolt (F-K) migration, F-X migration can accommodate lateral velocity changes.
- Works well in areas with strong velocity gradients (e.g., salt bodies, thrust belts).
3. Computationally Efficient
- Faster than Reverse Time Migration (RTM) because it operates in the frequency domain.
- Uses phase-shift extrapolation, reducing the need for expensive time-domain simulations.
4. Preserves Waveform Fidelity
- Maintains true amplitudes better than Kirchhoff migration.
- Produces cleaner images with fewer migration artifacts.
5. Works Well with Pre-Stack Data
- Can be applied to pre-stack data (e.g., common-offset gathers) for improved resolution.
Limitations:
1. Limited by Frequency Sampling
- Requires adequate frequency sampling—if too few frequencies are used, the image may suffer.
- Low-frequency noise can persist if not properly filtered.
2. Approximations in Wave Propagation
- Uses one-way wave equation, meaning it does not account for:
- Multi-path reflections (e.g., turning waves)
- Backscattered energy (which RTM handles better)
3. Struggles with Extreme Velocity Contrasts
- Sharp velocity boundaries (e.g., salt flanks) may cause artifacts.
- Less accurate than RTM in areas with strong velocity inversions.
4. Dip Limitations
- While better than Kirchhoff, it still has dip limitations (~60°–70°).
- RTM is superior for imaging near-vertical structures.
5. Memory and Storage Requirements
- Needs large RAM for 3D implementations due to frequency-domain storage.
- Less efficient than Kirchhoff migration for very large surveys.
Result of F-X Migration with AI
Phase Shift Migration
Concept:
- Frequency-Wavenumber domain operates by downward continuing wavefields using phase shifts in the Fourier domain.
- Applies a depth-dependent phase shift to each frequency-wavenumber component.
- Assumes vertically varying velocity (1D approximation) and lateral velocity variations are not properly handled.
Strengths:
1. High Accuracy for Vertically Varying Velocity
- PSM exactly handles vertical velocity variations (1D velocity, v(z)).
- It applies a phase shift in the f-k domain, which is mathematically precise for depth-dependent velocity models.
2. Computationally Efficient
- Operates in the Fourier domain, leveraging the Fast Fourier Transform (FFT), making it faster than time-domain methods like RTM.
- No need to solve partial differential equations (PDEs) iteratively.
3. No Dip Limitations
- Unlike Kirchhoff migration, PSM correctly handles steep dips (up to 90° in theory) because it doesn’t rely on ray tracing.
4. Free from Numerical Dispersion
- Since it works in the frequency domain, it avoids grid dispersion issues common in finite-difference methods.
5. Handles Multiples and Complex Wavefronts Well
- Better at imaging multiples and complex wavefronts than Kirchhoff migration (though not as good as RTM).
Limitations:
1. Limited to 1D Velocity (v(z))
- Cannot handle lateral velocity variations (v(x,z)) accurately.
- Requires Stolt stretch or other approximations for mild lateral variations.
2. Approximations for Strong Lateral Variations
- Extensions like Phase Shift Plus Interpolation (PSPI) or Split-Step Fourier help but introduce errors.
3. Artifacts in Complex Media
- Multiples, diffractions, and sharp contrasts may produce artifacts.
- Less accurate than RTM in highly complex velocity models (e.g., salt bodies).
4. Depth Imaging Only
- Primarily used for post-stack migration (time-to-depth conversion).
- Pre-stack PSM exists but is less common than RTM or Kirchhoff for pre-stack data.
5. Frequency Bandwidth Sensitivity
- Requires proper frequency filtering to avoid low-frequency noise.
- High-frequency loss can occur if the velocity model is too smooth.
Result of Phase Shift Migration with AI
Comparison of Phase Shift Migration result of AI with Industrial software
Wave Equation (Finite Difference) Migration
Concept:
- Wave equation migration (WEM) is a wavefield-based imaging method that accurately reconstructs subsurface structures by numerically simulating wave propagation.
- Unlike simpler methods (like Kirchhoff migration), WEM honors wave physics, making it ideal for complex geology.
Here's what happens step-by-step:
✅1. Input Data
- Stacked seismic section in time domain.
- A velocity model (in depth), typically smooth and derived from velocity analysis.
✅2. Conversion to Depth
Before migration, if your input data is in time, you usually:
- Convert the time axis to depth using the velocity model, or
- Use a time-to-depth mapping within the migration algorithm itself.
✅3. Wavefield Extrapolation
This is the core of wave-equation migration:
- The recorded seismic wavefield is back-propagated (reverse time or depth-stepping).
- A one-way wave equation (e.g., scalar wave equation, Helmholtz, or paraxial approximations like phase-shift, split-step, or finite difference) is used to extrapolate the wavefield downward in depth.
✅4. Imaging Condition
At each depth level:
- The extrapolated wavefield is correlated with the input data (or with itself in some approaches) to find the reflectivity.
- This gives an image of subsurface reflectors at their true depth positions.
Common imaging conditions:
- Zero-lag cross-correlation
- Kirchhoff integral form
- Energy stacking
✅5. Output
- A depth-domain image of the subsurface reflectivity.
- Reflectors are more accurately placed compared to time migration, especially in complex geology (dips, velocity contrasts, etc.).
✅6. Optional Enhancements
- Anti-aliasing: To avoid spatial aliasing when handling steep dips.
- Amplitude correction: Compensate for geometrical spreading and propagation effects.
- Frequency filtering: Control numerical dispersion and stability.
Strengths:
1. Accuracy in Complex Geology:
- Handles multi-pathing, steep dips (>90°), and overturning waves better than Kirchhoff migration.
- Preserves wavefront kinematics (phase and amplitude) more faithfully.
2. No High-Frequency Approximation:
- Unlike Kirchhoff (ray-based), WEM solves the full wave equation without assuming infinite frequency.
- Better for imaging low-contrast or gradational velocity boundaries.
3. Natural Handling of Multiples:
- Can image primaries and multiples coherently (with proper imaging conditions).
- Useful in subsalt or sub-basalt exploration.
4. Amplitude Preservation:
- Maintains relative amplitudes better than Kirchhoff, aiding AVO/AVA analysis.
- Critical for quantitative interpretation (QI).
5. Adaptability to Anisotropy/Attenuation:
- Extensions to anisotropic (TTI, VTI) and viscoacoustic/elastic media are straightforward.
Limitations:
1. Computational Cost:
- Much more expensive than Kirchhoff (especially 3D implementations).
- Memory-intensive due to wavefield storage (for reverse-time or phase-shift methods).
2. Velocity Model Sensitivity:
- Requires accurate velocity models. Errors cause defocusing or artifacts.
- Less tolerant to noise in velocity fields compared to Kirchhoff.
3. Low-Frequency Artifacts:
- Can generate "migration smiles" or low-frequency noise (requires post-migration filtering).
- Reverse-time migration (RTM) often needs Laplacian filtering to remove artifacts.
4. Limited Illumination Handling:
- Unlike beam/controlled-beam migration, WEM doesn’t explicitly compensate for illumination gaps.
- May underimage shadow zones (e.g., below salt).
5. Implementation Complexity:
- Requires careful handling of boundaries (PML/absorbing BCs).
- Stability issues (CFL condition) in finite-difference schemes.