The propagation of seismic waves may be described in the space-frequency domain by the Rayleigh-Sommerfeld convolution integral. The kernel of this integral is called a spatial wavelet and it embodies the physics and geometry of the propagation problem. The concepts of spatial convolution and spatial wavelet are simple and are similar to other topics studied by geophysicists. With a view to understanding these concepts, some aspects of spatial wavelets and their application to two-dimensional, zero-offset, acoustic seismic modelling were investigated. In studying the spatial wavelet, two topics in particular were examined: spatial aliasing and wavelet truncation. Spatial aliasing arises from the need to compute a discrete wavelet for implementation on a computer. This problem was solved by using an analytic expression for the spatial wavelet in the Fourier (wavenumber) domain. In the wavenumber domain the wavelet was windowed by a fourth order Butterworth operator, which removed aliasing. This technique is simple and flexible in its use. The second problem of wavelet truncation is due to the necessity of having a wavelet of finite length. A length limiting scheme based upon on the energy content of a wavelet was developed. It was argued that if that if a large portion of the wavelet energy was contained in a finite number of samples, then truncation at that sample would incur a minimal loss of information. Numerical experiments showed this to be true. The smallest length wavelet was found to depend on temporal frequency, medium velocity and extrapolation increment. The combined effects of these two solutions to the practical problem of computing a spatial wavelet resulted in two drawbacks. First, the wavelets provide modelling capabilities up to structural dips of 30 degrees. Second, there is a potential for instability due to recursive application of the wavelet. However, neither of these difficulties hampered the modelling of fairly complex structures. The spatial wavelet concept was applied to seismic modelling for media of varying complexity. Homogeneous velocity models were used to demonstrate diffraction evolution, dip limitations and imaging of curved structures. The quality of modelling was evaluated by migrating the modelled data to recover the time-image model of the reflection structure. Migrations of dipping and synform structures indicated that the modelled results were of a high calibre. Horizontally stratified velocity models were also examined for dipping and synform structures. Modelling these reflection structures showed that the introduction of a depth variable velocity profile has a tremendous influence on the synthetic seismic section. Again, migration proved that the quality of the data was excellent. Finally, the spatial wavelet algorithm was extended to the case of laterally varying velocity structures. The effects of space variant spatial convolution in the presence of a smoothed velocity field were examined. Smoothed velocity fields were computed by a simple weighted averaging procedure. The weighting function used was a decaying exponential whose decay rate determined the amount of smoothing. Seis-mograms computed for this case showed that the algorithm gave smoother and more continuous reflection signatures when the velocity field has been smoothed so that the largest lateral velocity gradient corresponded to the lower end of the temporal frequency band of the spatial wavelets. In this respect, the results are similar to those of geometric ray theory. Also, the travel times of these models compared favourably with those of ray tracings. / Science, Faculty of / Earth, Ocean and Atmospheric Sciences, Department of / Graduate

This thesis is concerned with the problem of estimating broadband acoustic impedance from normal incidence reflection seismograms. This topic is covered by following the linear inverse formalisms described by Parker (1977) and Oldenburg (1980). The measured seismogram is modelled as a convolution of subsurface reflectivity with a source wavelet. Then an appraisal of the seismogram is performed to obtain unique bandlimited reflectivity information. This bandlimited reflecitivity information is then utilized in two different construction algorithms which provide a broadband estimate of reflectivity; from which a broadband impedance function may be computed. The first construction method is a maximum entropy method which uses an autoregressive representation of a small portion of the reflectivity spectrum to predict spectral values outside that small portion. The second and most versatile construction method is the linear programming approach of Levy and Fullagar (1981) which utilizes the unique bandlimited spectral information obtained from an appraisal and provides a broadband reflectivity function which has a minimum 1( norm. Both methods have been tested on synthetic and real seismic data and have shown good success at recovering interpretable broadband impedance models. Errors in the data and the uniqueness of constructed reflectivity models play important roles in estimating the impedance function and in assessing its uniqueness. The Karhunen-Loeve transformation is discussed and applied on real data to stabilize the construction results in the presence of noise. The generally accepted idea that low frequency impedance information must be supplied from well log or velocity analyses because of the bandlimited nature of seismic data has been challenged. When accurate, bandlimited reflectivity information can be recovered from the seismic trace, then an interpretable, broadband impedance model may be recovered using the two construction algorithms presented in this thesis. / Science, Faculty of / Earth, Ocean and Atmospheric Sciences, Department of / Graduate

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<p>The resolution of near-surface seismic reflection data is often limited by attenuation and scattering in the shallow subsurface which reduces the high frequencies in the data. Compensating for attenuation and scattering, as well as removing the propagating source wavelet in a time-variant manner can be used to improve the resolution. Here we investigate continuous non-stationary iterative time-domain deconvolution (CNS-ITD), where the seismic wavelet is allowed to vary along the seismic trace. The propagating seismic wavelet is then a combination of the source wavelet and the effects of attenuation and scattering effects, and can be estimated in a data-driven manner by performing a Gabor decomposition of the data. For each Gabor window, the autocorrelation is estimated and windowed about zero lag to estimate the propagating wavelet. Using the matrix-vector equations, the estimated propagating wavelets are assigned to the related columns of a seismic wavelet matrix, and these are then interpolated to the time location where the maximum of the envelope of the trace occurs within the iterative time-domain deconvolution. Advantages of using this data-driven, time-varying approach include not requiring prior knowledge of the attenuation and scattering structure and allowing for the sparse estimation of the reflectivity within the iterative deconvolution. We first apply CNS-ITD to synthetic data with a time-varying attenuation, where the method successfully identified the reflectors and increased the resolution of the data. We then applied CNS-ITD to two observed shallow seismic reflection datasets where improved resolution was obtained. </p>

The Panola Mountain Research Watershed (PMRW) is a 41 ha forested watershed, located 25 km southeast of Atlanta, Georgia. Within that watershed is a 10 ha sub-catchment which contains the headwaters of the watersheds main stream, as well as two outcrops of Panola Granite (Burns et al. 2001). On the hill-slope below the northernmost outcrop, is a 20m long trench that has been excavated down to bedrock, a depth that ranges from 0.5-1.5m. In previous studies (Burns et al., 2001; Freer, et al., 2002), discharge through the overlying soils was measured along the trench in 2m sections across the full length of the trench. In those studies, it was assumed that the underlying bedrock was impermeable. However, Burns et al. (2003) showed that the riparian groundwater downslope from the trenched hillslope site was only 6 to 7 years old. The permeability of the Panola granite in question was proven by a recent Lithium- Bromide line tracer experiment -van Meerveld et al., in review. Due to the levels of bromide in the bedrock measured at the trench, it has become apparent that fluid is being lost to hidden hydraulic pathways, those probably being fractures. The objective of this thesis is to test the viability of using shallow seismic reflection to map out fracture zones at the trenched hillslope site, using GPR and shallow seismic refraction as supplemental techniques to verify the interpretation. Data from two seismic common shot point (CSP) surveys permitted an image of the soil/bedrock interface to be constructed, as well as enabling an acoustic velocity profile to be calculated for the area. This velocity profile is then used with data from five reflection surveys (one seismic and four radar) to create vertical profiles of the subsurface. Although the resulting SSR profile is less than optimum, higher amplitude arrivals related to structure were still able to be detected. From this profile, as well as those from the GPR survey, it can be concluded that the proposed hidden hydraulic pathways due indeed exist, and also that they can be accurately mapped out using the previously mentioned geophysical methods.

Thesis (Ph. D.)--University of Alberta, 2009. / Title from pdf file main screen (viewed on Dec. 1, 2009). "A thesis submitted to the Faculty of Graduate Studies and Research in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Geophysics, Department of Physics, University of Alberta." Includes bibliographical references.

Thesis (Ph.D.)--Memorial University of Newfoundland, 2001. / Bibliography: leaves 177-186. Also available online.

During the last decade, archaeologists have widely accepted the use of geophysical exploration techniques, including magnetic, resistivity and electromagnetic methods, for pre-excavation site assessment. Although researchers were quick to recognize the potential of seismic techniques to provide cross-sectional images of the subsurface, early feasibility studies concluded that seismic methods were inappropriate due to restricted resolving power and the relatively small-scale nature of archaeological features. Unfortunately, this self-fulfilling prophesy endures and has largely discouraged subsequent attempts to exploit seismic methods for archaeological reconnaissance. Meanwhile, however, seismic technology has been revolutionized in connection with engineering, groundwater and environmental applications. Attention to detail in developing both instrumentation and data acquisition techniques has yielded a many-fold improvement in seismic resolving power. In light of these advances, this dissertation re-examines the potential of reflection seismology for archaeological remote sensing. It is not the objective of this dissertation to deliver an unequivocal pronouncement on the ultimate utility of reflection seismology for the investigation of archaeological sites. Rather, the goal has been to establish a sound theoretical foundation for objective evaluation of the method's potential and future development. In particular, a thorough theoretical analysis of seismic detection and resolution yields practical performance and identifies frequency response characteristics associated with optimum resolution. Findings have guided subsequent adaptation, development and integration of seismic instrumentation, resulting in a prototype system for high-resolution seismic imaging of the shallow subsurface. Finally, to assess system performance and the suitability of optimum offset data acquisition techniques, a full-scale subsurface model has been constructed, allowing direct comparison between experimental soundings and known subsurface structure. Results demonstrate the potential of reflection seismology to resolve near-surface features on the scale of archaeological interest. Moreover, despite conventional wisdom that the groundpenetrating radar method possesses vastly superior resolving power, acquisition of coincident radar soundings demonstrates that the two techniques provide comparable resolution.

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