Harmonic Encoding for Seismic FWI Convergence
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Solution Overview
Problem
Current seismic data processing methods, particularly full-wavefield inversion (FWI), face significant computational challenges due to the time-consuming simulation of seismic data, especially when using explicit time-domain simulations and iterative methods, which are inefficient when dealing with multiple source gathers.
Innovation Solution
A deterministic method for selecting orthogonal encoding weights allows for simultaneous inversion of multiple encoded source gathers, using weight vectors generated from smoothly varying periodic functions or eigenvectors of a Laplacian matrix, enabling faster and more efficient seismic data processing by approximating the sequential FWI behavior.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Measurement precision
If multiple source gathers are inverted sequentially using conventional FWI, then measurement precision is maintained, but productivity deteriorates due to enormous computational cost and time
Solution Approach 1:
Multiple source gathers are merged into a single composite encoded gather by summing individual source gathers with different encoding weights. This allows simultaneous inversion of multiple sources in one computational pass, achieving speed-up by a factor of n while maintaining inversion accuracy through deterministic orthogonal encoding schemes
Solution Approach 2:
Encoding weights are applied as parameters to transform individual source gathers into encoded versions. Deterministic orthogonal encoding schemes (e.g., Hadamard, Fourier, Wavelet transforms) change the parameter representation of source data, enabling efficient simultaneous processing while preserving the ability to recover individual source information
2Productivity
If random encoding weights are used for simultaneous-source inversion, then productivity improves through speed-up, but measurement precision deteriorates because a single super shot does not contain enough information
Solution Approach 1:
Multiple realizations of encoded super shots are processed in continuous sequence, each with different deterministic encoding weights. The sum of these realizations progressively approximates the sequential FWI result, with convergence improving as the number of realizations increases. This continuous processing maintains information completeness while achieving speed-up
Solution Approach 2:
Deterministic orthogonal encoding schemes (such as Hadamard, Fourier, or Wavelet transforms) provide structured periodic patterns of encoding weights across multiple realizations. This periodic structure ensures that information from all sources is systematically distributed and recovered across the realization sequence, improving measurement precision compared to random encoding
3Measurement precision
If the number of realizations is increased to improve approximation accuracy, then measurement precision improves, but loss of time increases due to multiple simulation operations
Solution Approach 1:
Encoding weights are predetermined using deterministic orthogonal schemes (Hadamard, Fourier, Wavelet transforms) before the inversion process begins. This preliminary preparation eliminates the need for iterative optimization of weights during inversion, reducing computational overhead and time loss while maintaining approximation accuracy across multiple realizations
Data Source
AI summary
A deterministic method for selecting a set of encoding weights for simultaneous encoded-source inversion of seismic data that will cause the iterative inversion to converge faster than randomly chosen weights. The encoded individual source gathers are summed (83), forming a composite gather, and simulated in a single simulation operation. The invention creates multiple realizations of the simulation (84), each with its own encoding vector (82) whose components are the weights for the shots in the composite gather. The encoding vectors of the invention are required to be orthogonal (82), which condition cannot be satisfied by random weights, and in various embodiments of the invention are related to eigenvectors of a Laplacian matrix, sine or cosine functions, or Chebyshev nodes as given by the roots of Chebyshev polynomials. For non-fixed receiver geometry, an encoded mask (61) may be used to approximately account for non-listening receivers.