Efficient waveform tomography for lithospheric imaging: implications for realistic, two-dimensional acquisition geometries and low-frequency data

We provide a series of numerical experiments designed to test waveform tomography under (i) a reduction in the number of input data frequency components ('efficient' waveform tomography), (ii) sparse spatial subsampling of the input data and (iii) an increase in the minimum data frequency used. These results extend the waveform tomography results of a companion paper, using the same third-party, 2-D, wide-angle, synthetic viscoelastic seismic data, computed in a crustal geology model 250 km long and 40 km deep, with heterogeneous P-velocity, S-velocity, density and Q-factor structure. Accurate velocity models were obtained using efficient waveform tomography and only four carefully selected frequency components of the input data: 0.8, 1.7, 3.6 and 7.0 Hz. This strategy avoids the spectral redundancy present in 'full' waveform tomography, and yields results that are comparable with those in the companion paper for an 88 per cent decrease in total computational cost. Because we use acoustic waveform tomography, the results further justify the use of the acoustic wave equation in calculating P-wave velocity models from viscoelastic data. The effect of using sparse survey geometries with efficient waveform tomography were investigated for both increased receiver spacing, and increased source spacing. Sampling theory formally requires spatial sampling at maximum interval of one half-wavelength (2.5 km at 0.8 Hz): For data with receivers every 0.9 km (conforming to this criterion), artefacts in the tomographic images were still minimal when the source spacing was as large as 7.6 km (three times the theoretical maximum). Larger source spacings led to an unacceptable degradation of the results. When increasing the starting frequency, image quality was progressively degraded. Acceptable image quality within the central portion of the model was nevertheless achieved using starting frequencies up to 3.0 Hz. At 3.0 Hz the maximum theoretical sample interval is reduced to 0.67 km due to the decreased wavelengths; the available sources were spaced every 5.0 km (more than seven times the theoretical maximum), and receivers were spaced every 0.9 km (1.3 times the theoretical maximum). Higher starting frequencies than 3.0 Hz again led to unacceptable degradation of the results.