Source encoding for adjoint tomography

Abstract

We use a version of source encoding to facilitate the calculation of the gradient of a misfit function independent of the number of sources or receivers. The crosstalk-free method requires only two numerical simulations per iteration, namely, one source-encoded forward simulation and one source-encoded adjoint simulation. Importantly for practical applications, each source does not need to be recorded by all receivers. The method is implemented in two complementary ways. In the first approach the encoded forward and adjoint wavefields are run until they reach steady state, at which point they are ‘decoded’ to obtain their stationary parts. These stationary parts are combined to calculate the misfit gradient by summing their respective contributions. In the second approach the steady-state encoded forward and adjoint wavefields are convolved over a time period proportional to the inverse of the encoded frequency spacing. Using this strategy, the encoded forward and adjoint wavefields do not need to be decoded, nor is there a need to calculate or store intermediary stationary contributions to the gradient. We consider a wide variety of source-encoded misfit functions, including waveform differences, phase and amplitude measurements,‘double-difference’ phase and amplitude measurements, cross-correlation traveltime measurements, and a generic adjoint tomography misfit function. When measurements in specific time windows are involved in the construction of the source-encoded misfit function, as in adjoint tomography, the computational cost scales linearly with the number of seismic sources, because the necessary synthetic seismograms must be computed individually. In contrast, when using ‘super measurements’ based on source-encoded Fourier coefficients of entire observed and simulated seismograms, as in pure full waveform inversion, one iteration requires just two numerical simulations, independent of the number of sources and receivers. We illustrate the method based on examples from both earthquake and exploration seismology, highlighting inversion options and strategies involving frequency- and time-domain encoding, decoding (with more than 16 000 frequencies), encoded frequency randomization, encoding multiple frequencies per source, effects of noise, a variable number of receivers per event, various measurements and related misfit functions and attenuation.