Robert L. Nowack

Permeability dependence of seismic amplitudes

Can permeability be determined from seismic data? This question has been around since Maurice Biot, working for Shell in the 1950s, introduced the idea that seismic waves induce fluid flow in saturated rocks due to fluid-pressure equilibration between the peaks and troughs of a compressional wave (or due to grain accelerations in the case of a shear wave). Biot (1956) established a frequency-dependent analytical relation between permeability and seismic attenuation. However, laboratory, sonic log, crosswell, VSP, and surface seismic have all demonstrated that Biots predictions often greatly underestimate the measured levels of attenuation—dramatically so for the lower-frequency measurements. Yet, if an unresolved link truly exists between seismic amplitudes and permeability, the potential benefit to the oil industry is enormous. For this reason, the Department of Energy (DOE) brought together 15 participants from industry, national laboratories, and universities to concentrate for two days on whether permeability information is conceivably contained in and retrievable from seismic data. The present article represents much of the workshop discussion (which took place 5–6 December 2001 in Berkeley, California), but is not strictly limited to it. Not all connections between hydrological and seismic properties are considered. Three-dimensional seismic images and time-lapse seismic monitoring are routinely used by reservoir engineers in constructing and constraining their reservoir model. Such imaging applications of seismic surveys to hydrological modeling are not discussed. Furthermore, in fractured reservoirs it is reasonable to postulate that any locally determined seismic anisotropy defines a symmetry class for the geologic material that must also be satisfied by the permeability tensor. Neither are such material-symmetry constraints discussed. The focus here is only on whether the permeability of the rocks through which seismic waves propagate directly influences the decay of the wave amplitudes with distance. Key to addressing this question is an up-to-date discussion of the likely attenuation …

Calculation of Synthetic Seismograms with Gaussian Beams

In this paper, an overview of the calculation of synthetic seismograms using the Gaussian beam method is presented accompanied by some representative applications and new extensions of the method. Since caustics are a frequent occurrence in seismic wave propagation, modifications to ray theory are often necessary. In the Gaussian beam method, a summation of paraxial Gaussian beams is used to describe the propagation of high-frequency wave fields in smoothly varying inhomogeneous media. Since the beam components are always nonsingular, the method provides stable results over a range of beam parameters. The method has been shown, however, to perform better for some problems when different combinations of beam parameters are used. Nonetheless, with a better understanding of the method as well as new extensions, the summation of Gaussian beams will continue to be a useful tool for the modeling of high-frequency seismic waves in heterogeneous media.

Linearized rays, amplitude and inversion

In this paper, ray theoretical amplitudes and travel times are calculated in slightly perturbed velocity models using perturbation analysis. Also, test inversions using travel time and amplitude are computed. The pertubation method is tested using a 3-D velocity model for NORSAR having velocity variations up to 8.0 percent. The perturbed amplitudes are found to be in excellent agreement with the calculated ray amplitudes. Velocity inversions based on travel time and amplitude are next investigated. Perturbation analysis using linearized ray equations is efficiently used to compute amplitude derivatives with respect to model parameters. The trial linearized inversions use smaller velocity variations of 1.7 percent to avoid possible effects due to ray shift, even though the perturbation analysis is valid for larger variations. The trial 2-D inversion results show that linearized amplitude inversions are complementary and not redundant to travel time inversions, even in smoothly varying models.

Variations of P wave speeds in the mantle transition zone beneath the northern Philippine Sea

Using waveforms and travel times from deep earthquakes, we constructed 16 seismic profiles, each of which constrains the radial variation in Vp over a small area beneath the northern Philippine Sea. Taken together, the azimuthal coverage of these profiles also places tight bounds on the lateral extent of a region of anomalously high Vp (up to 3% faster than average Earth models) originally suggested by travel time tomography. Unlike travel time tomography, which relies heavily on arrival times of the direct P phase, we utilize the waveforms and move-out of later arrivals that mainly sample the mantle transition zone of interest. Our results identify three important characteristics of the northern Philippine Sea anomaly that are distinct from previous results. First, being approximately a subhorizontal, laterally uniform feature, the anomaly is localized beneath the northwestern corner of the Philippine Sea, within a region of approximately 500×500 km2 immediately east of the Ryukyu arc. Second, the anomaly is well constrained to occur in the lower portion of the transition zone, extending all the way down to the 660-km discontinuity. Third, the presence of such a distinct anomaly reduces the contrast in Vp across the 660-km discontinuity from approximately 6% to 3%. Such a configuration is consistent with the interpretation that the anomaly is caused by a remnant of subducted slab, as negative buoyancy should rest the slab just above the 660-km discontinuity where resistance to subduction is expected from a negative Clapeyron slope during the spinel—Mg-Fe-perovskite transition.

Correlation Migration Using Gaussian Beams of Scattered Teleseismic Body Waves

Correlation migration for structural imaging using Gaussian beams is described for the inversion of passively recorded teleseismic waves. Gaussian beam migration is based on an overcomplete frame-based representation of the seismic wave field and uses localized slant-stack windows of the data. Paraxial Gaussian beams are then utilized for the backpropagation of the seismic waves into the me- dium. The method can provide stable imaging of seismic data in smoothly varying background media where caustics and triplicated arrivals exist. We develop Gaussian beam migration for structural imaging, which allows for incident teleseismic waves from beneath the structure, using directly scattered or surface-reflected phases. The method is applied to synthetic data computed for a collisional-zone model, and the inversions show that the Gaussian beam migration can image structure using passive teleseismic data. The method is next applied to synthetic data that have been con- volved with an observed pulse from the 1993 Cascadia experiment. The autocorre- lation is computed and the autocorrelated data are migrated by using Gausian beams for direct P-to-S scattered waves and surface-reflected waves. Although the migra- tions of the autocorrelation data with the source pulse included have more noise than the migrations of the impulsive source data, the subduction zone structure can still be clearly identified in the migrated results without removal of the source pulses.

Gaussian beam migration for sparse common‐shot and common‐receiver data

We investigate the Gaussian beam migration of commonshot and common-receiver data. The imaging of commonshot data is useful for seismic data where the receiver coordinates are well sampled, but the source coordinates are less well sampled. By reciprocity, this approach can also be applied to common-receiver data, such as from OBS experiments where the source locations are dense but the receiver locations are sparse. Since Gaussian beam migration uses smoothed, localized windowing of the data, it can provide more stable results for the inversion of sparsely sampled data.

Focused gaussian beams for seismic imaging

The application of focused Gaussian beams is investigated for the seismic imaging of common-shot reflection data. The focusing of Gaussian beams away from the source and receiver aperture adds flexibility to beam imaging algorithms allowing for the narrowest portions of the beams to occur at the depth of a specific target structure. This minimizes the number of beams required to form an image at the target depth. The beam fronts at the beam-waists are also planar leading to more stable beam summations for imaging. To match with the surface data, a quadratic phase correction is required for the local slant-stacks of the data. Imaging using focused Gaussian beams is tested using a single shot gather for a model with 5 scatterers at different depths. The approach is then tested for a single shot gather from the Sigsbee2A model. In all cases, the beams can be focused to a particular target depth, but proper imaging still results for other depths as well.

Application of autoregressive extrapolation to seismic tomography

Seismic tomography used in the laboratory, as well as in the field, is strongly affected by limited and nonuniform ray coverage. A two-stage autoregres- sive extrapolation technique is proposed that can be used to extend the observed data and provide better tomographic images. The algorithm is based on the principle that the extrapolated data add minimal information to the existing data. The first stage of the extrapolation is to find the optimal prediction-error (PE) filter. The second stage is to use the PE filter to find the values of the missing data. The missing data are estimated to have the same spectrum as the observed data and are similar to maxi- mizing an entropy criterion. To test the method, synthetic tomography experiments for laboratory rock samples are used in which full ray coverage can be obtained. Autoregressive methods are then used to extrapolate the partial ray coverage and the tomographic results are compared with the full ray coverage case. The synthetic tests show that the autoregressive method can extrapolate known data to find missing data and can provide improved tomographic images. The autoregressive extrapolation is also tolerant to noise. Although the method was applied to a laboratory geometry where the ray coverage can be controlled, autoregressive methods may have impor- tant applications to tomography experiments in the field where complete ray coverage often cannot be easily realized. regressive methods to seismic tomography problems. We propose and implement a data extrapolation method based on Claerbouts approach for performing seismic data pre- diction beyond the measurement range. The idea of our al- gorithm is that the useful information in the known data is relevant and can be used to estimate the missing values by autoregressive extrapolation methods, and the estimated data only add minimum information to the known data. We apply this algorithm to synthetic arrival time data, with and without random noise added, and also to real laboratory experimental data. The results show that the proposed technique can ex- trapolate seismic travel-time data for seismic tomography, and the method is tolerant to random noise in the data.

Crustal Structure Beneath Taiwan Using Frequency-band Inversion of Receiver Function Waveforms

Abstract—Receiver function analysis is used to determine local crustal structure beneath Taiwan. We have performed preliminary data processing and polarization analysis for the selection of stations and events and to increase overall data quality. Receiver function analysis is then applied to data from the Taiwan Seismic Network to obtain radial and transverse receiver functions. Due to the limited azimuthal coverage, only the radial receiver functions are analyzed in terms of horizontally layered crustal structure for each station. In order to improve convergence of the receiver function inversion, frequency-band inversion (FBI) is implemented, in which an iterative inversion procedure with sequentially higher low-pass corner frequencies is used to stabilize the waveform inversion. Frequency-band inversion is applied to receiver functions at six stations of the Taiwan Seismic Network. Initial 20-layer crustal models are inverted for using prior tomographic results for the initial models. The resulting 20-1ayer models are then simplified to 4 to 5 layer models and input into an alternating depth and velocity frequency-band inversion. For the six stations investigated, the resulting simplified models provide an average estimate of 38 km for the Moho thickness surrounding the Central Range of Taiwan. Also, the individual station estimates compare well with the recent tomographic model of and the refraction results of Rau and Wu (1995) and the refraction results of Ma and Song (1997).

Tomography and the Herglotz-Wiechert Inverse Formulation

In this paper, linearized tomography and the Herglotz-Wiechert inverse formulation are compared. Tomographic inversions for 2-D or 3-D velocity structure use line integrals along rays and can be written in terms of Radon transforms. For radially concentric structures, Radon transforms are shown to reduce to Abel transforms. Therefore, for straight ray paths, the Abel transform of travel-time is a tomographic algorithm specialized to a one-dimensional radially concentric medium. The Herglotz-Wiechert formulation uses seismic travel-time data to invert for one-dimensional earth structure and is derived using exact ray trajectories by applying an Abel transform. This is of historical interest since it would imply that a specialized tomographic-like algorithm has been used in seismology since the early part of the century (seeHerglotz, 1907;Wiechert, 1910). Numerical examples are performed comparing the Herglotz-Wiechert algorithm and linearized tomography along straight rays. Since the Herglotz-Wiechert algorithm is applicable under specific conditions, (the absence of low velocity zones) to non-straight ray paths, the association with tomography may prove to be useful in assessing the uniqueness of tomographic results generalized to curved ray geometries.