U.S. Department of Energy
15 September 1998 to 14 September 1999
SEISMIC SURFACE WAVE TOMOGRAPHY OF WASTE SITES
Principal Investigator:††† ††††††††††† Leland Timothy Long
Institution:†††††††††††††††††††††††††††††††† Georgia Institute of Technology
††††††††††††††††††††††††††††††††††††††††††††††† School of Earth and Atmospheric Sciences
Project Number: ††††††††† G-35-W02
Grant Number: DE-FG07096ER14706
Grant Project Officers:† Nick Woodward
Project Duration†††††††††† 15 September 1998 to 14 September 1999
††††††††††††††††††††††††††††††††††† Extension to 14 September 2000
TABLE OF CONTENTS
††††††††††† The objective of the proposed work was to develop an analysis technique for surface-wave group-velocity tomography.† During three years of funding, we developed and tested tools, techniques and computer programs for the acquisition and analysis of surface-wave group-velocity data.† During the report period, the work concentrated on modifications and additions to the analysis technique.† In particular, we examined the application of minimum differences in observed travel times at adjacent stations as a constraint in the inversion process.† It turns out that a better procedure is to give uncertain values a low weight, rather than attempt to correct the value to something more correct based on adjacent values.† Using a combination of techniques, we successfully obtained tomographic images of the group velocity in 4 test areas.† In one area, the resulting dispersion curves were successfully inverted to obtain detailed images of the near-surface shear-wave velocity structure.† In other areas, the inversion of group velocity dispersion to shear-wave velocity structure became unstable and could not be automated for the computation of the thousands of pixels of a typical tomographic image.† The first stage in the planned continuation of this work is to develop a robust inversion technique for obtaining shear-wave structure from the dispersion curves.† At least six more data sets would be required to test and perfect the inversion and analysis programs.† A robust inversion method would make the tomographic technique ready for development as an analysis tool for general application in understanding and mapping waste sites.
The research objectives were to develop the computer programs and acquisition system for surface-wave group-velocity tomography and test these at three sites.
We measure group velocity of a selected frequency from the travel time of surface-wave motion between a source and an array of sensors. In this analysis ray theory is assumed. That is, we assume that the structures vary only slightly within distances of a wavelength. Elastic waves in general tend to average material properties with dimensions smaller than a wavelength and this averaging limits their ability to resolve detail.† Formally in ray theory, the travel time is computed from the integral of the slowness along a given, possibly curved, raypath
† ††††††††††††††††††††††††††††††††††††††††††† (1)
where †is the slowness (reciprocal of velocity) and is the line element along the raypath. A common approach in seismic tomography is to divide the medium into small blocks (pixels of the image) and estimate the slowness in each block from the observed travel times. We solve the linear discrete tomography problem by approximating equation (1) with constant slowness values in finite blocks.†† The discrete form of equation (1) is a matrix,
where t is the vector of observed travel times, s is the
slowness of the blocks, and is an †matrix of raypath
segments with †rays crossing the medium,
and blocks in the model.
The problem of tomography is to solve Eq. (2) for the unknown group slowness, the reciprocal of group velocity.† Most geophysical inverse applications, including this problem, lack sufficient data to guarantee uniform coverage and to remove all the singularities in the least-squares solution. In geophysical applications, noise in measured travel times can degrade the solution and can produce spurious velocity anomalies. In order to obtain a solution to Eq. (2), we use iteration to find the damped least squares solution. We start with a guess, an a priori solution, s(k), and find the new solution, s(k+1), using the alternate form for damped and weighted least squares (Menke, 1984, p. 55, Eq. 3.40),
where, Wb is weight assigned to each block, in this study the number of times a block is sampled by a ray, Wt is the covariance matrix for the observed times, and e is the damping coefficient, which is chosen by trial and error to give the greatest resolution without causing instability in the inversion.† The solution to Eq. (3) becomes the next guess and the iteration is repeated until the guess converges to the solution.† The weights, Wb and Wt, are generally assumed to be diagonal.† The inverse to †is difficult to compute for large data sets and we approximate this term by its diagonal. This approximation is justified when most of the elements of are zero and tend to cancel out the off-diagonal elements.† This is generally the case in tomography because each ray samples only a small subset of blocks comprising the model. The damped and weighted least squares solution of Eq. (3) is computationally a rigorous derivation and improvement to the formulas used in the simultaneous iterative reconstruction technique (SIRT) (see Lo and Inderwiesen, 1994, p. 36, Eq. (46)).† In SIRT, the differences between observed and predicted travel times are back projected to obtain average slowness perturbations to be used to update the latest slowness model.† We have developed programs to compute the improved version of SIRT.
So far, four new data sets have been obtained for testing our analysis techniques.† Two are in the soft soils of the Coastal Plain sediments, one is at the Georgia Tech Cobb County test site, and the fourth augments the ORNL data set in Tennessee.† Only the Coastal Plain site at Hamburgh State Park will be described.† The Hamburgh State Park test site is located in east central Georgia, USA. The near-surface materials are sands and clays characteristic of the Coastal Plain sediments of Cretaceous age.†
Figure 1.† Seismic traces from a single shot located near the upper right-hand (northeast) corner of the rectangle. The traces are recorded on the western border and extend from north (1) to south (16).
The test geometry selected for the surface-wave tomography experiment is a square 29.36m (96ft) on a side. The sensors were placed 2m (6ft) apart along the north (top) and west sides. The shots were symmetrically placed 2m (6ft) apart along the south and the east sides. The combination of 16 shots and sensors on opposite sides provides a density of ray coverage from 10 to 40 paths crossing imaging pixels of 1.0m in size.† This is sufficiently dense to allow pixels that are half the seismometer spacing in the interior of the rectangle.†† We used an eight-pound hammer with a trigger switch attached to the hammerís head. A typical recording of the response to an eight-pound hammer (Fig. 1) shows the moveout with distance expected for a shot point in one corner. The dominant amplitudes correspond to the surface waves, principally the fundamental mode of Rayleigh waves, recorded by1.0 Hz vertical component seismometers. We recorded 16 channels at 6250sps for 640ms.† The data acquisition was accomplished with a portable computer with an analog to digital converter card and acquisition programs written at as part of the initial program.† Additional programs were written to convert the acquired data to standard SEGY formats.
In the multiple filter technique, seismic data are filtered in the frequency domain with a narrow-band Gaussian filter. The group velocity for frequencies at the center of the narrow-band filter is estimated from the arrival time of the peak instantaneous amplitude.† The center frequency of the narrow-band filter is applied successively at 1 Hz increments.† In order to obtain the instantaneous amplitudes for each filtered trace, we compute the analytic signal, which is a complex signal with its real part defined by the actual trace and its imaginary part defined by the Hilbert transform of the real part.† The instantaneous amplitude of the output is the magnitude of the complex trace in the time domain. The results of multiple filtering are shown in Fig. 2 for 16 Hz. The trace-to-trace variation in arrival times of the peak instantaneous amplitudes for a given frequency indicates the group velocity anomaly along the ray-paths of the particular source-receiver geometry.† Along a line, the arrival of the wave group can shift across phases as they do in Fig. 2.
The amplitude peaks may be shifted by noise.† Also, when waves are focused or scattered by anomalous structures and arrive at anomalous times, they may introduce shifts in the times of peak amplitudes. In order to eliminate the arrival times of peaks corresponding to interfering waves, such as the shear wave, the P-wave, an acoustic air wave or other sources of noises, we limit the picking of peaks to times close to the expected group arrival time.† Furthermore, we applied an accept/reject condition to eliminate those arrivals that fall outside acceptable arrival times or that were inconsistent with neighboring arrival times. A stacking procedure was used to estimate the group velocity and arrival times, or they were determined interactively through direct observation of the filtered traces.† For some traces, particularly those that are at higher frequencies and at greater distances, no obvious peak may exist in the instantaneous amplitude and these spurious values are also eliminated with the accept/reject condition.† To accomplish this analysis we wrote an interactive visual program using MATLAB analysis package.† This program has been automated and tested with our current data for eventual conversion to executable form in FORTRAN, C, or Visual BASIC.
The times corresponding to peaks in the amplitudes are the observed arrival times that provide the basic data for the tomographic image at each frequency.† However, these arrival times must first be corrected for group delays related to ground coupling of the instrument, instrument response, and source function.† We have determined that the instrument and its coupling to the ground can impart significant delays to the group arrival time. If left uncorrected, the velocities can be in error by 10 to 25 percent, particularly for frequencies near the geophoneís natural frequency. While the instrument group delay can be determined exactly from the instrumentís impulse response, the coupling of the instrument to the ground and the source function delays depend on the condition of the ground. The coupling of the geophone generally resonates at frequencies one to two orders of magnitude higher than the geophone's natural period, but on soft ground it can be lower and significant for the some frequencies of interest.† Average group delays that include all source geophone effects can be measured directly from field data if data from a refraction line are available. Refraction line data, from an area of suspected uniform structure, are processed for group delays with the multiple filter technique, thus including any group delays possibly introduced by the analysis technique. As the surface waves cross the survey area, the group arrival times at each frequency can be extrapolated back to the source and a delay time determined. These delay times are then subtracted from the observed times prior to further processing.
In general, the group arrival times are sufficiently precise to give a tomographic image. However, the uncertainty of picking a group arrival time depends inversely on the width of the filter.† Because the objective in this study is to determine the frequency dependence of group velocities, we use a narrow filter.† For a narrow filter, the uncertainties of the picks can be large compared to the time one would expect for a surface wave to propagate from an adjacent station. In order to improve the quality of the image by reducing the scatter in the group arrival times, we considered adding a constraint.† The constraint is that the difference between the times of arrival at adjacent stations must be similar to the difference in travel times along the expected propagation paths with an uncertainty defined by the uncertainty in the direction of propagation in an inhomogeneous media.† A constrained and weighted least-squares reduction could then be used to determine the optimal arrival times by combining the amplitude picks and expected time differences between traces that follow similar paths.† In this way, the different lines are tied together and random deviations in picks of arrival times are suppressed.† The details of this reduction depend on the geometry, the number of similar paths included in the reduction and number of sides of the study area along which data were obtained.† Application of this constraint significantly reduced the noise in the tomographic inversion.† However, because the constraint assumes an average velocity, it introduces long wavelength artifacts in some data sets.† We obtained better results by utilizing damping and the uncertainties of the picks in the modified version of SIRT used to generate the tomographic image.
We obtained group velocity images at 1.0 Hz increments from 15 to 50 Hz (see Fig. 3 for selected frequencies).† The images are 30 by 30 pixels, each slightly less than 1.0m2, corresponding to 900 unknowns covering an area of 894m2. These images show a pattern of anomalies that change gradually with frequency. In Fig. 3 the group velocity ranges from a low of 100 to a high of 300 m/s.† For the lower frequencies, there is a distinct difference between the higher velocities in the north and the lower velocities in the south. For the intermediate frequencies, the anomalies are less distinct at an average of 225 m/s. Above 35Hz the waves predominately sample the top two meters. The images begin to show a pattern that is related to surface features but may also include noise at the highest frequencies. For example, the lowest velocities at the higher frequency are along the north edge where the area was at the edge of a young pine forest as opposed to a previously plowed field and the near-surface soil would contain more humus accumulations.†††
Once the tomographic images are developed, dispersion curves for any pixel may be generated. Usable dispersion curves were obtained for square areas with sides on the order of 1.0m, a single pixel.† For modeling shear-wave velocities at each pixel, we used a model with 12 one-meter thick layers over a half space. In general, the model parameters include layer thickness, shear-wave velocity, P-wave velocity and density. In this study we solve only for shear-wave velocity, fixing the layer thickness and using assumed values for density and Poisson's ratio. The inverse method outlined in Kocaoglu and Long (1993) was used to solve for the model. Given an initial estimated model, the initial estimate of the group velocity dispersion curve can be computed directly from the dispersion relation for surface waves and the model parameters (Herrmann, 1987).† Then, by using the first two terms of a Taylor's expansion, corrections to the initial estimate can be obtained by iteration until convergence is achieved at values within the estimated error of the measurements. The Jacobian matrix is computed numerically by using a finite difference approximation. For all iterations, singular value decomposition is used in the generalized inverse for stability.† Dispersion curves could be interpreted reliably for variations in velocity to a depth of 8.0m.†
The group velocity images suggest that the principal structure in this area is two-dimensional with an east-west strike. The resulting structure (Fig. 4) indicates that the high velocities that are most obvious at 20 Hz (Fig. 3) are caused by shallow high-velocity structure. In this study area, the shear-wave velocity varied from 200 m/s at the surface to 450 m/s at 4 meters depth.†† However, the image shows considerable detail suggesting a possible thrust fault or edge of a buried ditch. There is a distinct 2.0m vertical displacement of the velocity structure across a line near 18 meters. The resolution was sufficient to resolve anomalous features to within 1.0 m, the size of the resolution pixel and half the geophone spacing. The character of the anomaly with depth suggests a thrust fault or, alternatively, the existence of an area near 10.0m that has been disturbed to a depth of 4.0m.† In this case, surface-wave group-velocity tomography has effectively imaged a geologic structure.
Most surface-wave methods now used in the evaluation of near-surface velocity structure are based on measurements of phase velocity.† Phase velocity can be measured directly and assigned an error, whereas group velocity arrival time picking requires indirect measurements and greater uncertainty.† When measured in the field at discrete frequencies, generated by single-frequency signal generators, the surface wave techniques are generally referred to as SASW (Spectral Analysis of Surface Waves) tests in the engineering literature (Stokoe et al., 1989).† The SASW tests utilize two (or more) sensors over the test zone and measure phase velocity in the frequency domain from the phase shift between two or more sensors.† In order to examine the spatial variation in the near-surface shear-wave velocities, the SASW test would have to be repeated at many points over an area.† Because the precision of the phase shift measurement is proportional to sensor spacing, the spatial resolution is limited.† Improvements in SASW are currently being developed by students of Glenn Rix, Georgia Institute of Technology, School of Civil and Environmental Engineering. For example, by adding attenuation to the inversion (Lai, 1998) and by using optimal and predictive spatial filtering of array data to remove errors from refraction and higher mode interference (Zywicki, 1999).† However, these improvements do not remove the limitation on spatial resolution inherent in SASW techniques, nor do they address the influence of gradients on propagation of surface waves.†
The most significant similar research is by Dr. Choon Park and associates at the Kansas Geological Survey (for example, Park et al., 1999).† Their technique (referred to as the multichannel analysis of surface wave technique, MASW) is equivalent to using the SASW method along an extended refraction line.† An advantage of this technique is their ability to use standard refraction line analysis programs.† The disadvantage relative to our tomography is that it gives structures in two dimensions only.† Phase velocity is mapped as a function of frequency and distance along the line.† Then phase velocity dispersion curves are generated as a function of position along the refraction line.† Each dispersion curve is inverted for shear-wave structure.† As in our work, the inversion of the dispersion curve has proven to be unstable or unreliable for routine application.† Observations of their data suggest that changes in shallow structure can affect the interpretation of deeper structure, when these structures should be independent.† This unusual correlation can come from constraints in the inversion used to maintain stability.†† One significant observation from their data is the emergence of higher modes as a function of distance.† They suggest that differences in the attenuation properties of shallow soils cause some modes to attenuate rapidly and others to emerge as the dominant signal.† We have not recognized this as a problem with surface-wave group-velocity tomography, perhaps because our arrivals are at similar distances or because we restrict interpretation to arrival times consistent with the fundamental mode.† The emergence of higher modes as a dominant phase may be explained by the existence of strong gradients which we propose to study.† The finite difference simulations can determine whether the emergence of higher modes is a function of attenuation of fundamental modes or through conversion from fundamental mode to higher mode along the propagation path.†
Relevance, Impact and Technology Transfer
During the course of the work, we have developed computer programs to carry out the unique aspects of surface-wave group-velocity tomography.† We believe that our computer programs and preliminary results demonstrate that useful data can be obtained using surface-wave group-velocity tomography.††
Fundamental to relevance is the question as to why we should use surface waves.† Surface waves are uniquely suited for the estimation of near-surface shear-wave velocities. They are usually the largest amplitude waves generated by a surface impact, their velocity is determined primarily by the shear-wave velocity of materials in a depth range of ľ wavelength, and their dispersion properties allow separation of different wavelengths for interpretation of velocity as a function of depth.† In seismic tomography, waves crossing a study area are measured on its boundary in order to image the interior. The resolution of the image is limited by wavelength and imaging technique.† Surface-wave group-velocity tomography has been used in seismology for over 30 years to study global crustal structure and recently to study regional structure and sedimentary basins (Kafka and Reiter, 1987; Kocaoglu and Long, 1993). In this project, we have applied the tomographic inversion of surface-wave velocities to areas with dimensions appropriate for near-surface structures that are often encountered in environmental problems.
A second fundamental question concerns why we should use tomography.† The application of surface-wave tomography to near-surface structures has advantages and problems. The problems are introduced by the complexities of the structure and by the indirect and computationally intensive techniques that are inherent in surface wave interpretation. Structures in the near surface often do not satisfy the layering of most analytical models developed originally for a predominantly layered earth. If the structure varies significantly within a wavelength of the surface wave, a conversion of wave motion to higher modes of propagation can interfere with interpretation of the fundamental mode. The velocity contrasts may also be great at shallow depths, grading quickly from loose sands to unweathered granite. These conditions lead to wave modeling and resolution problems. However, the interpretation techniques presented in this paper were not significantly affected by converted waves and the computational complexity was easily within the capabilities of portable computers. The advantages of tomography are sufficient to warrant the use of surface waves in waste site evaluations.† For example, an image of shallow structure can be obtained from the periphery of sites with limited access.† In areas where trenches were used to dump wastes, the area of the trench will be revealed as an anomalous velocity. In areas where seepage from a waste site is controlled by structure, the structures may be defined and used as a guide for drilling and sampling.† The sensitivity of surface wave velocity to fluid content could eventually allow surface-wave tomography to track fluid movement with time.
Surface-wave group-velocity tomography has the potential of becoming a powerful near-surface imaging technique for shear-wave velocity structure.† We are proposing to develop a new approach to the inversion of dispersion curves that is theoretically correct and can include the effects of a velocity gradient.† We expect to demonstrate whether there is a significant difference between propagation in a velocity gradient and propagation in a layered media.† We expect that the greatest impact of a gradient model will be in the existence and character of the higher modes.† The higher modes, which correspond generally to higher frequencies, in a layered model are equivalent to multiply reflected shear waves.† A gradient strongly reduces these reflections and could significantly modify the appearance of higher modes.†
Shear waves are very sensitive to the existence of fluids in soils when the fluids are at or near to saturation.† Hence, the mapping of time variations in the depth to the water table in soils may become possible without having to drill many wells to get single point values.† These could include detection of the existence of dense fluids as contaminates.
To date we have limited our studies to shallow and small areas.† With a larger source, the use of lower frequencies could increase the depth of effective imaging and allow applications with larger and deeper targets.
Surface-wave group-velocity tomography should be compared to those of SASW. However, the tomography should provide structures in three dimensions, not just average structure with depth.† Tomography should improve the spatial resolution over that possible with multiple SASW tests and provide it with significantly less field effort.†
Shear-wave velocity, or equivalently shear modulus, is a very important parameter in the design of foundations for structures.† Additional evidence for the validity of surface-wave group-velocity tomography could help support the use of SASW and related surface wave techniques in providing estimates of shear modulus for construction.†
This project is the shear-wave interpretation portion of a more complete analysis package for group-velocity surface-wave tomography.† We will continue to refine the acquisition and analysis techniques and prepare these for dissemination and application to special projects. The approach we are proposing for inversion of the dispersion curves is unique and may have applications appropriate for other techniques such as SASW and velocity measurements using sensor arrays.
We are currently investigating various avenues to distribute the surface-wave group-velocity tomography analysis techniques developed under this contract.† For example, we are discussing the possibility of developing a commercial package in cooperation with major instrument manufacturer. The analysis package would be marketed as part of a seismic acquisition system and we would jointly run training workshops for equipment operators.† A commercial package would require a robust dispersion inversion program, such as proposed in this project.† Alternatively, we could make the analysis programs available through Georgia Tech and provide support and documentation, perhaps through workshops.
The industry is generally reluctant to adopt and use (i.e. spend real dollars on) an unproven technique like surface-wave group-velocity tomography.† Adoption of this technique will come only after a number of case studies are successfully demonstrated and after convenient (i.e. user friendly) packages are available.† Hence, a natural extension of the proposed work would be to seek new "problem" areas to image and to use as demonstration data sets.† Such areas could help to further develop the analysis programs and improve their ability to handle a wide variation in field conditions, as well as build confidence in the capabilities of the technique.† In particular, the application to specific critical problem should be evaluated.† Such problems could include assessment of permafrost degradation, flow of dense fluids, and a study of the relation between age of burial and ease of detection.† The decrease in detection capability for older structures is expected because the disturbance of a burial will heal with time and the rate of this healing and its impact of seismic velocities is not well understood.† There exist additional interpretation problems to evaluate, such as the impact of trees of various size on the passage of surface waves.††
The project results, which met or exceed these objectives are reported in papers, expanded abstracts, and talks (see * references in Bibliography).† An overview of surface-wave group-velocity tomography is given in our paper submitted to Journal of Environmental and Exploration Geophysics (JEEG).† As described in the JEEG paper and summarized below, we have developed recording and analysis techniques to obtain the group velocity dispersion curves for a portion of a study area with a resolution theoretically limited only by the wavelengths of the surface waves.†
††††††††††† Leland Timothy Long, Principal Investigator
††††††††††† Dr. Argun Kocaoglu, post doctoral assistant.† Dr. Kocaoglu worked on the data processing techniques and program development related to the time-frequency analysis of the seismograms and generation of synthetic dispersion curves.
††††††††††† Dr. Jeffrey Martin, post-doctoral assistant.† Dr. Martin assisted in programming and testing techniques related to applying constraints to the inversion and to applying corrections for the instrument-induced group-velocity delay.
Xiuqi Chen, graduate research assistant.† Xiuqi Chen assisted in the field data acquisition and programs to convert data to different formats.
††††††††††† Thomas Collins, undergraduate assistant.† Mr. Collins assisted with the data acquisition instruments.
††††††††††† Brook Miller, undergraduate assistant.† Mr. Miller programmed a data acquisition system for multi-channel data acquisition.
(see papers submitted below)
Martin, J. T., T. Kubota, and L. T. Long, Imaging near-surface buried structure with high-resolution surface-wave group-velocity tomography, Abstract submitted to 2000 International Conference on Image Processing (ICIP), September 10-13, 2000, Vancouver, BC, Canada.
Long, L. T., Kocaoglu, A. H., and Martin, J., Shallow S-wave structure can be interpreted from surface-wave group-velocity tomography, in Proceedings of the Symposium on the Application of Geophysics to Engineering and Environmental Problems, Environmental and Engineering Geophysical Society, February, 2000†† (SAGEEP00)
Long, L.T., A. Kocaoglu, W.E. Doll, X.Q. Chen, J. Martin. Surface-Wave Group-Velocity Tomography for shallow structures at a waste site, SEG Expanded Abstract, Annual Meeting, Houston, November 1-3, 1999.
Long, L.T., and Kocaoglu, A., (1999). Surface-Wave Group-Velocity Tomography for Shallow Structures, in Proceedings of the Symposium on the Application of Geophysics to Engineering and Environmental Problems, Environmental and Engineering Geophysical Society, March, 1999††† (SAGEEP99)
Long, L.T.(1999). Seismic Surface Wave Tomography at Waste Sites, Research Note in: Fast Times, The EEGS Newsletter, February.
Long, L.T., and A. Kocaoglu (submitted September 1999) Surface-Wave Group-Velocity Tomography for Shallow Structures, Journal of Environmental and Engineering Geophysics.
Long, L.T. and Argun Kocaoglu, 1999.† A tomographic inversion method for near-surface structure, (Abstract, Eastern Section Seismological Society of America, Annual meeting October 16-20, 1999, Memphis, TN.
Long, Leland Timothy, Surface-Wave Group-Velocity Tomography, (a power point presentation), Earth and Atmospheric Sciences Departmental Seminar, spring 1999.
† †Field data were obtained to supplement existing seismic data in a waste area of the Oak Ridge National Laboratory site.† The supplemental data covered two squares with 30 meter sides within an area of existing tomographic data.
The objectives and scope of the contract (D-FG07-96ER14706) remains the same. However, some changes are needed to effect a more robust solution technique.† We originally assumed that the conversion of dispersion curves to shear-wave structure could be accomplished using well-known and "tested" techniques.† Our assumptions were wrong for some structures.† These techniques were found to work for some of our areas, but became unstable in others, particularly those with low-velocity zones at depth. The instability in these cases is related to the numerical difficulties in distinguishing fundamental and higher modes.† While techniques exist to sort out the fundamental and higher modes, they are not practical for the automatic inversion of many dispersion curves (over 256 in each of our current models).† Also, the existing inversion programs are based on constant velocity layers; an approximation that may not be appropriate for the strong depth-dependence in velocities typical of soils and their transition to unweathered rock.† Programs to exactly model a gradient velocity structure do not currently exist (except as an approximation with a sequence of thin layers). Our objective for future work is thus to develop a robust and accurate inversion method specifically for the velocity gradients of a soil.† We propose to base this inversion on exact solutions from finite difference simulations of dispersion.† Our updated objective is to develop an analysis package for data acquisition, data reduction, and data interpretation in terms of shear-wave structure that can be used to evaluate the three-dimensional structure and time variations in structure of near-surface soils.† In the process of developing the analysis program, we propose in a project continuation to obtain and process data from 6 additional areas in order to test the robustness and accuracy of the interpretation.†
Two tasks are proposed.† We will simultaneously 1) develop a robust inverse based of finite difference simulation of surface wave propagation in soils with velocity gradients and 2) obtain new data at 6 test sites to use in further testing and improving the analysis technique.† Our approach is to continue to use the developed techniques, but to make continued improvements, specifically in the area of inversion for shear-wave structure.††
1) Most FD programs are based on constant velocity layers.† We will use programs, such as those used by Long and Liow (1990) that include a velocity gradient and boundary conditions for a free surface.† Alternatively, we will reformulate the pseudo spectral FD programs developed by Kocaoglu (1995) for study of the influence of topography on surface-wave propagation.† The pseudo spectral FD method is more accurate at higher frequencies that conventional FD formulation.† With an operating FD program, we will take the synthetic seismograms and use our group-velocity analysis program to compute the corresponding dispersion relation.† This model will then be changed systematically for each orthogonal function in order to determine the elements in the Jacobian matrix.† The orthogonal functions will be designed to have uniform significance among the frequencies for which reliable group velocities are obtained.
2) Experience with a variety of field conditions can be most important in revealing problems in modeling techniques.† We propose to obtain data at 6 more test sites.† We will attempt to provide analysis that cover larger areas.† Hence, we propose to add a larger weight-drop source for recording data.† The larger source will allow lower frequencies to be included in the analysis and allow inversion for deeper structures.† We hope that at least one of these can be on the Savannah River Project site at which collaborative data are available.† Other test sites will be chosen to demonstrate the influence of different soils and rocks.†† A second site will be chosen to demonstrate time variations in fluid saturation.
The specific problem to be investigated in the future is how to develop a robust and accurate solution for shear-wave structure from surface-wave group-velocity dispersion curves and to incorporate it into an analysis program for surface-wave group-velocity tomography for three-dimensional shear-wave structure.† By robust, we mean a solution that is stable with perturbations of the velocity structure.† By accurate, we mean one that simulates and represents a gradient model for shear-wave structures.
An accurate inverse model for soil requires a forward model that includes the gradients in velocity that are typical in soils.† Existing forward models for computing dispersion curves from the shear-wave structure represent soils as layers of constant-velocity.† In this form, the modeling of dispersion becomes numerically more intense and unstable as the number of layers increases, making a reasonable approximation to a gradient numerically difficult.† This proposal is to introduce a forward modeling technique based on finite difference methods that accurately model a velocity gradient and accurately simulate waveforms from all modes excited by a surface source.† The finite difference technique is based on the equation of motion in a media and can be written to include a velocity gradient.† The introduction of a complete equation of motion solution, instead of its approximation with the wave equation, will improve the solution in three ways.† First, the analysis will be based on time traces of surface waves that are processed in the same way as the field data.† This will cancel out any errors introduced in the data processing techniques.† Second, by using waveforms computed for distances consistent with the field data the influence of distant dependent emergence of higher modes, perhaps related to a velocity gradient, will be suppressed. Third, only the modes and frequencies introduced by a surface source will enter into the solution.† The forward models for layered media provide solutions for all modes, even those that carry little energy and are not observed in field data.† It is the detection of these higher modes, those that exist in structures with low-velocity zones and that have velocities similar to the velocities of the fundamental mode, that cause instability in most attempts to automate the inversion for shear-wave structure.
The typical shear-wave versus depth relation for a soil and its transition to weathered rock is a gradient.† Where sharp boundaries exist, such as with unweathered rocks, buried cement slabs or excavated materials, the contact can be approximated with a sharp gradient, perhaps more accurately than a gradient can be approximated with layered media.† In the Oak Ridge National Laboratory data set for the surface-wave group-velocity tomography, data from seismic refraction lines were obtained for correlation with our tomography results.† The composite interpretation of all these data (Fig 5.) when plotted as a Log-Log plot show a straight line that suggests that the P-wave velocity satisfies a power law relation with depth.† This data also shows a velocity jump to unweathered rock at depths of 20 to 40 meters.† Others have documented similar relations.† In general the exponential increase in velocity holds universally, but the absolute value of the velocities and the gradient varies strongly with soil type, age, composition, and other locally controlled factors.††
The conventional technique for the inversion of dispersion curves for shear-wave structure starts with a guess in the form of a layered structure.† The group (or phase) velocity is a non-linear function of the layer velocities.† Therefore, the first two terms in a Taylorís series expansion are used to compute corrections for the initial guess of layer velocities.† Because the equations are nonlinear, the process is repeated until a stable solution is found.† In our analysis, we start with a gradient velocity structure.† The perturbations will not be layers.† Instead, we will maintain a gradient by using continuous orthogonal functions to represent the perturbations in velocity as a function of depth.† In our solution the elements of the Jacobian matrix are the derivatives (at each frequency) of the group velocity with respect to the orthogonal function, instead of a layer velocity as in the conventional method.† If the initial guess is a gradient and is close to the final solution, then the elements of the Jacobian matrix will change slowly and the derivative equations would only need to be computed once.† This is the technique we propose to develop.
Lo, Tien-when, and Inderwiesen, P., 1994, Fundamentals of seismic tomography: Geophysical Monograph Series, No. 6, Society of Exploration Geophysicists, Tulsa, Ok.
Kafka, A. L., and Reiter, E. C., 1987, Dispersion of Rg waves in southeastern Maine: Evidence for lateral anisotropy in the shallow crust: Bulletin of the Seismological Society of America, 77, 235-238.
Herrmann, R. B., 1987. Computer Programs in Seismology, Vol. 4, Surface Waves Inversion: Saint Louis University, Missouri.
Kocaoglu, A. H., 1995.† A new method for modeling surface wave propagation in heterogeneous media, Ph.D. Thesis, Georgia Institute of Technology, Atlanta, 137 pp.
Kocaoglu, A. H., and Long, L. T., 1993, Tomographic inversion of Rg wave group velocities for regional near-surface velocity structure: Journal of Geophysical Research, v. 98, B4, p. 6579-6587.
Lai, Carlo Giovanni, 1998.† Simultaneous Inversion of Rayleigh Phase Velocity and Attenuation for Near-Surface Site Characterization, Ph.D. Thesis, Georgia Institute of Technology, Atlanta, GA 30332, 370pp.
Long, L.T., and J‑S. Liow, (1990).† A transparent boundary for finite difference wave simulation, Geophysics, No. 2, 201-208.
Menke, W., 1984, Geophysical data analysis: discrete inverse theory: Academic Press, Inc., New York, New York.
Park, C.B., Miller, R. D., and Xia, J., 1999.† Multi-channel analysis of surface waves, Geophysics, 64, 800-808.
Stokoe, K.H. II, Rix, G. J., and Nazarian, S., 1989, In situ seismic testing with surface waves: Proceedings, 12th International Conference on Soil Mechanics and Foundation Engineering, Rio De Janeiro, 12-18 August, p. 331-334
†Zywicki, Daren J., 1999.† Engineering Analysis of Seismic Surface Waves, Ph.D. Thesis, Georgia Institute of Technology, Atlanta, GA 30332, 328pp.