Modal separation, mapping and inverting three-component VSP data
In offset vertical seismic profiling (VSP), the large variation in angles of incidence at the geophone's location and the presence of considerable amounts of compressional (P) and shear (S) waves in the propagating wavefield necessitate the use of three-component geophones for full wavefield understanding. In this study, the vertical and radial components are used to recover the full amplitudes of P and vertical-shear (SV) waves, to map the P and SV waves to their respective reflection points, and to perform a reflectivity inversion on the converted-wave map in an attempt to recover shear-wave velocity information. Algorithms are derived and applied to two different data sets; a synthetic data set and a real three-component data set from the Rolling Hills area of southern Alberta, Canada. First the data are prepared for modal separation by applying a series of processes including rotation of horizontal components, trace equalization, downgoing wavefield removal, deconvolution and time-variant gain.
For the separation of P and S waves, filter coefficients are derived by considering the particle displacement of downgoing P and SV waves at the geophone's location. The separation process requires an estimate of P and S velocities along the borehole and uses several other simplifying assumptions. The filter coefficients are not all stable for vertical slownesses larger than 1/Vs and are tapered to zero by a cosine function to account for this instability. The filtering process is attempted in the f-k and localized p-t domains, and the results obtained through each method are compared. It is found that the results obtained through the p-t decomposition method do not contain the 'ringing' and the smearing of the wavelet in time and in depth that result when the f-k transform is used as a means of slowness decomposition.
The total P- and SV-wave sections ehused SP are then used to map the upgoing wavefields to their respective reflection points. The synthetic results show that the mapping algorithms map the P or SV reflections within 7% of their correct offset locations and within 4% of their correct time position. The SV-wave map of the synthetic data is used to test a velocity inversion algorithm that attempts to recover the S-wave velocities from the mapped traces. The inversion is found to give results that are within 1% of the actual velocities for the synthetic data case. For the real VSP data case, however, the inversion must be given constraints to perform adequately. The wavelet effect, caused by input data that is not spiked appropriately, is present in the inversion results of both the synthetic and the real data sets.