Complex wavelet based full waveform inversion method and system for processing seismic data

A computing device for imaging a formation into a subsurface, the computing device comprising:

• an interface configured to receive an initial velocity model V of the subsurface, and to receive recorded seismic data d related to the subsurface, wherein the recorded seismic data d represents a recorded wavefield of physical seismic waves detected by sensors;
• and
• a processor connected to the interface and configured to,
• generate synthetic data u representing a synthetic wavefield related to the subsurface, using the initial velocity model V and a source signature of a source S, which is used to generate waves into the subsurface;
• transform the recorded wavefield and the synthetic wavefield, with a complex wavelet transform, into complex wavelet transformed recorded data d′ and complex wavelet transformed synthetic data u′, respectively, to separate kinematics of the recorded and synthetic wavefields from dynamics of the recorded and synthetic wavefields trough phase and amplitude components;
• update the initial velocity model V using a full waveform inversion, FWI, to generate an updated velocity model, based on a cost function J which depends on the complex wavelet transformed recorded data d′ and the complex wavelet transformed synthetic data u′, wherein the updating comprises:
• (i) simulating a physical time-reversal of wave propagation by computing an adjoint wavefield for an adjoint source S′ at a receiver position where the recorded seismic data d is recorded, wherein the adjoint wavefield is computed as a derivative of the cost function J with respect to phase or amplitude to isolate kinematic errors;
• (ii) calculating a cross-correlation between a forward propagated source wavefield and the computed adjoint wavefield to generate a gradient;
• and
• (iii) updating the initial velocity model V using the gradient to mitigate a non-linearity related to cross-contamination of kinematic and dynamic effects in the velocity model;
• and
• generate an image of the formation in the subsurface based on the updated velocity model, wherein the formation is used to locate natural resources.

A non-transitory computer readable medium including computer executable instructions, wherein the instructions, when executed by a processor, implement a method for imaging a formation in a subsurface, the medium comprising instructions for:

• receiving an initial velocity model V of the subsurface;
• receiving recorded seismic data d related to the subsurface, wherein the recorded seismic data d represents a recorded wavefield of physical seismic waves detected by sensors;
• generating synthetic data u representing a synthetic wavefield related to the subsurface, using the initial velocity model V and a source signature of a source S, which is used to generate waves into the subsurface;
• transforming the recorded wavefield and the synthetic wavefield, with a complex wavelet transform, into complex wavelet transformed recorded data d′ and complex wavelet transformed synthetic data u′, respectively, to separate kinematics of the recorded and synthetic wavefields from dynamics of the recorded and synthetic wavefields trough phase and amplitude components;
• updating the initial velocity model V using a full waveform inversion, FWI, to generate an updated velocity model, based on a cost function J which depends on the complex wavelet transformed recorded data d′ and complex wavelet transformed synthetic data u′, wherein the updating comprises:
• (i) simulating a physical time-reversal of wave propagation by computing an adjoint wavefield for an adjoint source S′ at a receiver position where the recorded seismic data d is recorded, wherein the adjoint wavefield is computed as a derivative of the cost function J with respect to phase or amplitude to isolate kinematic errors;
• (ii) calculating a cross-correlation between a forward propagated source wavefield and the computed adjoint wavefield to generate a gradient;
• and
• (iii) updating the initial velocity model V using the gradient to mitigate a non-linearity related to cross-contamination of kinematic and dynamic effects in the velocity model;
• and
• generating an image of the formation in the subsurface based on the updated velocity model, wherein the formation is used to locate natural resources.