On the scalability of seismic waveform inversion: From ultrasonic experiments to global-scale tomography

Seismic waveform inversion endeavors to extract high-resolution subsurface models from full seismic records by using numerical solutions of the full forward wave equation. In the last two decades, waveform inversion has been successfully applied to both active and passive seismic experiments, and has demonstrated superb resolution power over a wide-range of applications. Waveform inversion is still computationally challenging, and is well known to be a strongly non-linear and ill-conditioned inverse problem. Passive and active waveform inversions have largely been developed independently, although both originate from the work of Lailly (1983) and Tarantola (1984). In this presentation, we review a suite of past results from both active and passive source waveform inversions, and attempt to illustrate similarities and shared challenges between them. In active seismics, waveform inversion has been applied to i) ultrasonic breast cancer data to image targets of a few tens of centimeters on a side, ii) cross-well exploration data using frequencies between hundreds and a few thousands of Hz to image targets of the order of hundreds of meters on a side, iii) surface seismic for near-surface engineering problems using frequencies of tens to a few hundreds of Hz to image targets of the order of hundreds of meters to several kilometers, and iv) hydrocarbon exploration and crustal imaging using frequencies of a few, to a few tens of Hz for targets tens of kilometers on a side. In contrast, waveform inversion from passive source data uses much lower frequencies (less than 1 Hz), and images much larger target areas to retrieve iv) crustal scale structures of the order of hundreds of kilometers using data periods of one to several tens of seconds, v) upper-mantle structure on regional scales thousands of kilometers on a side using data periods of ten to a few hundred seconds, and vi) the whole-earth inversions (of order 10,000 km on a side), using a similar frequency range. Our main emphasis is on the scalability of waveform inversion in these examples: The applicable spatial size of waveform inversion varies over eight orders of magnitude (tens of cm to thousands of km), and the frequency range of available seismic data is over nine order of magnitude (mHz - MHz). A dimensionless measure of the "scale" is the approximate number of wavelengths between source and receiver intervals N_λ . This non-dimensional parameter controls the non-linearity of waveform inversion, which increases strongly with N_λ . Remarkably, for most past successful waveform inversions 5<N_λ <100, regardless of physical scale. This observation suggests that many inversion techniques and procedures developed originally for either active or passive waveform inversion may be more widely applicable. Moreover further scrutiny and comparison of these waveform inversion results will likely provide valuable and more general insights into the non-uniqueness and the non-linearity of waveform inversion.