POCS Operator for Stable Seismic Inversion
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Solution Overview
Problem
Full-Wavefield Inversion in seismic data processing often leads to unstable simulations due to unbounded growth of numerical solutions, particularly when model updates violate stability constraints, causing a 'blow-up' and requiring smaller step sizes, which slows down the inversion process.
Innovation Solution
The method employs Projection Onto Convex Sets (POCS) to convert unstable rock physics models into stable ones by defining a projection operator that transforms matrices into the nearest positive semi-definite matrix, while also satisfying additional constraints related to rock-physics relationships, ensuring stability and convergence.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Productivity
If a large step size is used in model update, then the inversion process converges faster, but the simulation becomes unstable and blows up
Solution Approach 1:
The patent applies preliminary action by projecting the model update onto the convex set of stable models before using it in the inversion process. This preliminary projection ensures that any model update, no matter how large the step size, will result in a stable model that does not blow up during simulation. The projection operator P(m) transforms an arbitrary model m into the nearest stable model, preventing instability before it can cause problems.
Solution Approach 2:
The patent uses an intermediary approach by introducing a projection operator as a mediator between the model update step and the simulation process. This projection operator acts as a filter that removes unstable components from model updates, allowing large step sizes to be used without causing simulation blow-up. The intermediary projection ensures that only stable models proceed to the simulation stage.
2Reliability
If a small step size is used to ensure stability, then the simulation remains stable, but the inversion process slows down
Solution Approach 1:
The patent applies preliminary action by projecting the model update onto the convex set of stable models before using it in the inversion process. This preliminary projection ensures that any model update, no matter how large the step size, will result in a stable model that does not blow up during simulation. The projection operator P(m) transforms an arbitrary model m into the nearest stable model, preventing instability before it can cause problems.
Solution Approach 2:
The patent changes the parameter space by transforming models into a constrained subspace defined by the convex set of stable models. By projecting updates onto this constrained space, the effective step size in the stable subspace can be larger without causing blow-up, as the projection automatically adjusts the model parameters to maintain stability while preserving the direction of the update.
3Reliability
If projection onto convex sets is applied to enforce stability constraints, then simulation stability is improved, but computational complexity increases
Solution Approach 1:
The patent replaces complex mechanical constraint enforcement methods (such as penalty functions or Lagrange multipliers that require iterative adjustment) with a direct mathematical projection operation. The projection operator P(m) provides a closed-form solution that directly computes the nearest stable model without requiring iterative optimization, significantly reducing computational complexity while maintaining stability constraints.
Solution Approach 2:
The patent extracts the stability constraint enforcement from the main inversion loop by using a separate projection operator that can be applied independently. This extraction allows stability to be enforced as a separate computational step rather than being embedded in the complex iterative optimization process, simplifying the overall algorithm structure and reducing computational burden.
Data Source
AI summary
Method for stabilizing the updated model (13) in iterative seismic data inversion so that the model-simulated data for the next iteration does not “blow up.” A Projection Onto Convex Sets (“POCS”) operator is defined that converts the matrix corresponding to the model to a positive semi-definite matrix. The stability projection operator may be looped with physical constraint projection operators until the model converges (15). The resulting stable and constrained model is then used to simulate seismic data in the next cycle of the outer iteration loop (16).