ALBERT

Notes: Many methods for the analysis of long-period and broad-band S-waveforms depend on a representation of the seismic wavefield in terms of a sum of modes of a reference structure. In addition it is frequently assumed that the propagation of the modal contributions from source to receiver may be regarded as independent. This assumption may not be warranted if there is significant heterogeneity in seismic properties along the propagation path.The interaction between modal components in different styles of heterogeneity model for the upper mantle is examined using a coupled mode propagation technique (Kennett 1984) which allows direct construction of reflection and transmission matrices with full allowance for intermode interactions for 2-D heterogeneity structures.The first type of structure considered has been proposed to explain amplitude and traveltime anomalies in body wave studies of upper mantle phases. This heterogeneity has an amplitude of about 1 per cent and is distributed with a horizontal scale of around 300-400km and a vertical scale which increases from 70km in the uppermost mantle to 200 km at 900 km depth. Horizontally travelling S-waves are hardly affected by this class of heterogeneity for frequencies less than 0.07 Hz.The second heterogeneity model was based on the WEPL3 model proposed by Nolet (1990) from waveform inversion for the structure under the NARS array in western Europe. The heterogeneity reaches 5 per cent deviation from the reference model PREMC in organized regions 700 km or more in length. For this structure surface wave modes with group velocity below 4.2kms-’ can be regarded as propagating independently up to 0.020Hz. The body wave group of modes with higher group velocity succumbs to significant interaction above 0.040 Hz. The frequency limit for largely independent propagation for the body wave group of modes can be extended to about 0.05 Hz for a velocity model with up to 1 per cent additional variability superimposed on WEPL3. Such a composite heterogeneity model would be consistent with both body wave and surface wave behaviour.The errors introduced into the analysis methods by ignoring mode interactions above these frequency limits will depend on the distribution of energy across the modes imposed by the source, and the criterion used for waveform matching between observed and theoretical seismograms.Theoretical seismograms for the heterogeneity structures based on WEPL3 including full allowance for intermode coupling show a distinct phase shift for the fundamental mode when compared with the corresponding calculations for the reference model PREMC: as is indeed observed at the NARS stations. The parts of the seismograms which show the largest influence from the presence of lateral heterogeneity are those which depend on interference phenomena, such as Su. The disruption of the phase patterns can have a profound influence on the appearance of the waveforms. The presence of small-scale heterogeneity has very little influence on the seismograms below 0.035 Hz but becomes more important as the frequency increases, especially for the body wave phases.

Notes: A seismic survey was carried out on a tidal flat in the SW-Netherlands in order to determine shear-wave velocities in sediments by means of higher-mode Rayleigh waves. The dispersion properties of these Rayleigh waves were measured in the 2-D amplitude spectrum–or f, k-spectrum–and resulted in phase velocities for six different modes as a function of frequency (5–30 Hz). These observed phase velocities were inverted for a nine-layer model for the shear-wave velocity to a depth of 50 m.

Notes: When inverting large matrices, iterative techniques are necessary because of their speed and low memory requirements, as opposed to singular value decomposition (SVD). Recently, there have been attempts to obtain information on the quality of the solutions calculated using conjugate gradient (CG) methods such as LSQR. The purpose of this note is to comment on the paper titled “Estimation of resolution and covariance for large matrix inversions’ by Zhang & McMechan (1995), who extend Paige and Saunders’ LSQR algorithm to obtain an orthonormal basis used to approximate resolution and covariance. We show that for larger problems, where the number of orthogonal vectors is several orders of magnitude smaller than the number of model parameters, the vectors obtained do not adequately span the range of the model space. We use a synthetic borehole experiment to illustrate the differences between the singular value spectrum obtained through the more complete method of SVD and the Ritz value spectrum that results from a simple extension of LSQR, We also present a trivial numerical example to illustrate the differences between Zhang & McMechan's approximate resolution matrix and the true resolution.