Seismic tomography is an imaging technique which creates maps of subsurface elastic properties such as P/S wave velocity, density and attenuation, based on observed seismograms and use of sophisticated inversion algorithms. Amongst different acquisition geometries, seismic cross-hole tomography has a special position in geophysical surveys with many applications in hydrocarbons, coal and other minerals exploration and engineering purposes investigations related to constructions. Main goal of these studies is obtaining precise information about the earth structure (layers structure, impedance of layers, faults and fractures) or anomalies (objects, pipes, voids).
Traveltime tomography is a conventional approach to convert special phase of waveform travletimes (such as P or S wave arrivals) to corresponding parameters. Low computational effort is needed to perform traveltime tomography, but the results suffer from the lack of high resolution. Seismic waveform tomography is an efficient tool for high resolution imaging of complex geological structures and has been widely used by researchers in the field of exploration seismology. As waveform tomography exploits waveforms, in addition to traveltimes, it has superior resolution comparing to traveltime tomography but its computational complexities have limited its everyday use in real world applications.
In this study we focus on application of waveform tomography in an engineering purpose seismic cross-hole study. Our approach relies on solution of acoustic wave equation in frequency domain and minimizing residual of calculated wavefield and observed seismograms. Frequency domain approach lets simultaneous sources modeling and implementing frequency dependent absorption mechanisms. This approach leads to a large system of equations. To solve the large system of equations sparse direct solvers can be used. The mixed-grid finite-difference used to discretize continuous second order hyperbolic acoustic wave equation. Although elastic modeling is more the realistic and near to observed data, most researchers prefer to use acoustic wave equation instead of elastic one due to lower computational costs. Instead, we pre-process the observed data to increase comparability of observations and modeling. These pre-processing include suppressing phases cannot be explained by acoustic modeling such as S waves or Rayleigh waves or scaling seismograms to take into account amplitude vs. offset effects in acoustic and elastic cases. Waveform tomography is very a nonlinear problem with a very rugged cost function. To overcome this nonlinearity, we solve the problem using hierarchical approaches. We start inversion from low frequency components, where the cost function is smoother, and then proceed to higher components. Lower frequency inversion results have been used as initial velocity model for higher frequency inversion.
A synthetic example has been used to test the performance of the algorithm in the absence and presence of noise. As the results show the performance of current waveform tomography algorithm decreases in case of noisy data, which implies the importance of denoising before inversion and/or employing regularization. Another strategy which helps to control noise issue is simultaneous inversion of frequency components in different groups, as showed in real data example. Lastly a real cross-hole dataset acquired for engineering purposes has been studied. The traveltime tomography result is used as starting model for waveform tomography. The results of waveform tomography are in agreement with downhole measurements.