Seismic Simulation Temporal Dispersion Correction via Frequency Domain Resampling
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Solution Overview
Problem
Seismic simulators using finite difference algorithms suffer from temporal dispersion errors, which degrade the accuracy of Reverse Time Depth Migration (RTM) and Full Waveform Inversion (FWI) results, especially when higher-order approximations are not corrected, leading to incorrect frequency propagation and reduced accuracy in hydrocarbon prospecting.
Innovation Solution
A method that corrects temporal numerical dispersion by performing a Fourier transform on seismic data, resampling in the frequency domain to map incorrect frequencies to correct ones, and then performing an inverse Fourier transform, allowing for accurate seismic simulation with larger time steps while maintaining the accuracy of simulations with smaller time steps.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Productivity
If 2nd order time stepping is used in finite difference algorithms, then computational efficiency is improved and fewer resources are required, but temporal dispersion artifacts are introduced that reduce accuracy
Solution Approach 1:
The patent introduces a frequency-dependent scaling factor as an intermediary element that corrects the temporal dispersion artifacts. This scaling factor acts as a mediator between the computationally efficient 2nd order time stepping and the accurate representation of wave propagation, allowing the use of lower-order approximations without sacrificing accuracy. The scaling factor is applied in the frequency domain to compensate for the numerical dispersion introduced by the finite difference approximation.
Solution Approach 2:
The patent changes the parameter of time step size (Δt) while compensating for the resulting temporal dispersion through frequency-dependent scaling. Instead of using smaller time steps to maintain accuracy (which would reduce productivity), the method allows larger time steps and corrects the resulting dispersion artifacts through parameter adjustment in the frequency domain. This parameter change approach resolves the contradiction by decoupling time step size from accuracy requirements.
2Manufacturing precision
If higher-order approximations are used to reduce temporal dispersion errors, then accuracy is improved, but computational cost and resource requirements increase
Solution Approach 1:
The frequency-dependent scaling factor serves as an intermediary that provides the accuracy benefits of higher-order approximations without requiring the computational complexity. Instead of implementing computationally expensive higher-order finite difference schemes, the patent uses this scaling factor to achieve similar accuracy improvements at lower computational cost.
Solution Approach 2:
The patent creates a corrected version of the seismic data by applying the frequency-dependent scaling factor to the results obtained from 2nd order time stepping. This copying approach allows the use of simple, efficient algorithms while producing results that match the accuracy of more complex higher-order methods, effectively copying the beneficial outcome without the computational burden.
3Productivity
If larger time steps are used in simulations, then productivity is improved, but temporal numerical dispersion increases reducing measurement precision
Solution Approach 1:
The patent changes the time step parameter to larger values for improved productivity, then compensates for the resulting frequency propagation errors through frequency-dependent scaling. This parameter change strategy allows the use of larger time steps while maintaining measurement precision by adjusting the frequency domain representation of the seismic data.
Solution Approach 2:
The frequency-dependent scaling factor acts as a feedback mechanism that corrects the temporal dispersion introduced by larger time steps. The scaling factor is determined based on the frequency content and time step size, providing a corrective feedback that restores accuracy to the simulation results even when using computationally efficient larger time steps.
Applied Scientific Principles
This section explains which scientific principles are used to turn an abstract innovation direction into a practical engineering solution.
Function Achieved in This Case
This method effectively removes or introduces temporal numerical dispersion, improving the accuracy of seismic simulations and inversion results, reducing computational costs, and enhancing the precision of hydrocarbon prospecting by ensuring all frequency components propagate at the same speed, thus improving the quality of RTM and FWI outputs.
Implementation Method 1
performing a Fourier transform in time on (i) the simulated or (ii) the measured seismic data, then resampling the transformed seismic data in frequency domain
Data Source
AI summary
Method for correcting seismic simulations, RTM, and FWI for temporal dispersion due to temporal finite difference methods in which time derivatives are approximated to a specified order of approximation. Computer-simulated seismic data (51) are transformed from time domain to frequency domain (52), and then resampled using a mapping relationship that maps, in the frequency domain, to a frequency at which the time derivative exhibits no temporal dispersion (53), or to a frequency at which the time derivative exhibits a specified different order of temporal dispersion. Alternatively, measured seismic data from a field survey (61) may have temporal dispersion of a given order introduced, by a similar technique, to match the order of approximation used to generate simulated data which are to be compared to the measured data.