Crosshole seismic waveform tomography – II. Resolution analysis

In an accompanying paper, we used waveform tomography to obtain a velocity model between two boreholes from a real crosshole seismic experiment. As for all inversions of geophysical data, it is important to make an assessment of the final model, to determine which parts of the model are well-resolved and can confidently be used for geological interpretation. In this paper we use checkerboard tests to provide a quantitative estimate of the performance of the inversion and the reliability of the final velocity model. We use the output from the checkerboard tests to determine resolvability across the velocity model. Such tests can act as good guides for designing appropriate inversion strategies. Here we discovered that, by including both reference-model and smoothing constraints in initial inversions, and then relaxing the smoothing constraint for later inversions, an optimum velocity image was obtained. Additionally, we noticed that the performance of the inversion was dependent on a relationship between velocity perturbation and checkerboard grid-size: larger velocity perturbations were better-resolved when the grid-size was also increased. Our results suggest that model assessment is an essential step prior to interpreting features in waveform tomographic images.     

Regional 3-D heterogeneities by waveform inversion – application to the Atlantic area

The waveform inversion method described in Woodhouse & Dziewonski (1984) was modified to retrieve regional scale 3-D heterogeneities by using the minor arc part of seismograms. The lateral heterogeneities are expanded horizontally into blocks (10°× 10°) and radially into Legendre polynomials up to order 3 (0–670 km), and thus the results show much fine details of lateral variation than previous global scale studies. We assumed that the heterogeneities produce the perturbation of eigenfrequencies which are the minor arc average of local eigenfrequency shift. We applied the method to the upper mantle beneath the Atlantic Ocean and its environments. Care was taken about the weighting of the data set. We found that the fit of each seismogram became better when the weighting of each seismogram is proportional to the inverse of initial data residuals. Resolution is good in the triangular region surrounded by South America, Europe, and North America. Resolution is not good in the South Atlantic because of the poor path coverage. Depth resolution is not clear, because of the use of Legendre polynomials, though the results suggest a broad half-width of the order of 200 km or more. We found some similarities between previous global studies and our results. For example, low velocities beneath the East Pacific Rise, Chile Rise and Azores triple junction and a high velocity Canadian shield are obtained. However, there are also differences; the high-velocity zone beneath the Brazilian shield at shallow depth is not a prominent feature in this study. Instead, we found a somewhat unexpected feature near the Romanche and Vema fracture zones where shallow positive anomalies exist. Smoothed results calculated by the spherical harmonic expansion are also shown for the purpose of comparison with global studies.     

Two efficient algorithms for iterative linearized inversion of seismic waveform data

We present two equivalent algorithms for iterative linearized waveform inversion for 3-D Earth structure with respect to an arbitrary 3-D starting model; one is a matrix formulation, and the second is a wavefield formulation. Both algorithms require the computation of accurate synthetic seismograms, but neither requires that any particular method be used to compute the synthetics. The matrix formulation is equivalent to our previously published algorithm (Hara, Tsuboi & Geller 1991), but requires less than 10 per cent of the CPU time of the previous algorithm. The wavefield algorithm is equivalent to that of Tarantola (1986) and Mora (1987), but appears to be substantially more efficient.     

Non-linear optimization for seismic traveltime tomography

This paper presents a non-linear algorithmic approach for seismic traveltime. It is based on large-scale optimization using non-linear least-squares and trust-region methods. These methods provide a natural way to stabilize algorithms based on Newton's iteration for non-linear minimization. They also correspond to an alternative (and often more efficient) view of the Levenberg-Marquardt method. Numerical experience on synthetic data and on real borehole-to-borehole problems are presented. In particular, results produced by the new algorithm are compared with those of Ivansson (1985) for the Kråkemåla experiment.     

Inversion of seismic attributes for velocity and attenuation structure

We have developed an inversion formuialion for velocity and attenuation structure using seismic attributes, including envelope amplitude, instantaneous frequency and arrival times of selected seismic phases. We refer to this approach as AFT inversion for amplitude, (instantaneous) frequency and time. Complex trace analysis is used to extract the different seismic attributes. The instantaneous frequency data are converted to t * using a matching procedure that approximately removes the effects of the source spectra. To invert for structure, ray-perturbation methods are used to compute the sensitivity of the seismic attributes to variations in the model. An iterative inversion procedure is then performed from smooth to less smooth models that progressively incorporates the shorter-wavelength components of the model. To illustrate the method, seismic attributes are extracted from seismic-refraction data of the Ouachita PASSCAL experiment and used to invert for shallow crustal velocity and attenuation structure. Although amplitude data are sensitive to model roughness, the inverted velocity and attenuation models were required by the data to maintain a relatively smooth character. The amplitude and t * data were needed, along with the traveltimes, at each step of the inversion in order to fit all the seismic attributes at the final iteration.     

Simultaneous velocity and interface tomography of normal-incidence and wide-aperture seismic traveltime data

A tomographic inversion technique that inverts traveltimes to obtain a model of the subsurface in terms of velocities and interfaces is presented. It uses a combination of refraction, wide-angle reflection and normal-incidence data, it simultaneously inverts for velocities and interface depths, and it is able to quantify the errors and trade-offs in the final model. The technique uses an iterative linearized approach to the non-linear traveltime inversion problem. The subsurface is represented as a set of layers separated by interfaces, across which the velocity may be discontinuous. Within each layer the velocity varies in two dimensions and has a continuous first derivative. Rays are traced in this medium using a technique based on ray perturbation theory, and two-point ray tracing is avoided by interpolating the traveltimes to the receivers from a roughly equidistant fan of rays. The calculated traveltimes are inverted by simultaneously minimizing the misfit between the data and calculated traveltimes, and the roughness of the model. This 'smoothing regularization' stabilizes the solution of the inverse problem. In practice, the first iterations are performed with a high level of smoothing. As the inversion proceeds, the level of smoothing is gradually reduced until the traveltime residual is at the estimated level of noise in the data. At this point, a minimum-feature solution is obtained, which should contain only those features discernible over the noise.
The technique is tested on a synthetic data set, demonstrating its accuracy and stability and also illustrating the desirability of including a large number of different ray types in an inversion.