Virtual Source Method Research Articles

Modeling of vibration, sound, and stresses using the virtual source approach

Analytical solutions of acoustical, vibration, or stress problems are available only for some simple structural systems, e.g., a rectangular simply supported plate or a parallelepipedic cavity. An usual way of finding a solution where complex systems are concerned is to apply field discretization in conjunction with some numerical methods such as FE or BE. Yet the use of analytical models is desirable for its simplicity and better physical understanding of phenomena concerned. On top of this, analytical models give the possibility of assessing quantities like energy flow which are linked to higher spatial derivatives, the latter being difficult to model numerically. The method of virtual sources enables one to obtain analytical solutions of systems of rather simple but non-trivial geometry. A key advantage of the method is the full control over the computation error. The method consists in applying a layer of virtual sources to a simple mother system of known analytical solution. These sources are adjusted in such a way as to produce particular boundary conditions on a target part of the mother system. The target part can be given a complex geometry which cannot be directly treated analytically. The paper will be accompanied by a number of examples which illustrate the approach.

Seismic response of a single and a set of filled joints of viscoelastic deformational behaviour

Rock joints are often filled with weak medium, for example, saturated clay or sand, of viscoelastic nature. Their effects on wave propagation can be modelled as displacement and stress discontinuity conditions. The viscoelastic behaviour of the filled joint can be described by either the Kelvin or the Maxwell models. The analytical solutions for wave propagation across a single joint are derived in this paper by accounting for the incident angle, the non-dimensional joint stiffness, the non-dimensional joint viscosity and the acoustic impedance ratio of the filled joint. It is shown that the viscoelastic behaviour results in dissipation of wave energy and frequency dependence of the reflection and transmission coefficients. Based on curve fitting of the experimental data of P-wave propagation across a single joint filled with saturated sand, both the Kelvin and Maxwell models are found to reproduce the behaviour of the filled joint, in terms of the amplitude and frequency contents. Then, wave transmission across a filled joint set is studied with the virtual wave source method and the scattering matrix method, where multiple wave reflections among joints are taken into account. It is shown that the non-dimensional joint spacing and the number of joints have significant effects on the transmission coefficients.

Controlled-source interferometric redatuming by crosscorrelation and multidimensional deconvolution in elastic media

Various researchers have shown that accurate redatuming of controlled seismic sources to downhole receiver locations can be achieved without requiring a velocity model. By placing receivers in a horizontal or deviated well and turning them into virtual sources, accurate images can be obtained even below a complex near-subsurface. Examples include controlled-source interferometry and the virtual-source method, both based on crosscorrelated signals at two downhole receiver locations, stacked over source locations at the surface. Because the required redatuming operators are taken directly from the data, even multiple scattered waveforms can be focused at the virtual-source location, and accurate redatuming can be achieved. To reach such precision in a solid earth, representations for elastic wave propagation that require multicomponent sources and receivers must be implemented. Wavefield decomposition prior to crosscorrelation allows us to enforce virtual sources to radiate only downward or only upward. Virtual-source focusing and undesired multiples from the overburden can be diagnosed with the interferometric point-spread function (PSF), which can be obtained directly from the data if an array of subsurface receivers is deployed. The quality of retrieved responses can be improved by filtering with the inverse of the PSF, a methodology referred to as multidimensional deconvolution.

Seismic refraction interferometry with a semblance analysis on the crosscorrelation gather

Crosscorrelating wavefields recorded at two receivers to produce data as if one receiver was a source is commonly referred to as seismic interferometry, or the virtual source method. An artifact in seismic interferometry related to critically refracted waves allowed us to estimate the velocity in the refracting layer. In addition, we devised a new semblance analysis on the crosscorrelation of reflection and refraction energy to robustly estimate the depth and velocity of the slow layer, tested with a numerical example and field data from the Boise Hydrogeophysical Research Site.

Normally incident wave propagation across a joint set with the virtual wave source method

This paper presents an analytical study on normally incident wave propagation across a joint set. Combined with the displacement discontinuity method (DDM), the virtual wave source method (VWSM) is adopted to take account of multiple wave reflections among joints. After theoretical formulation, a comparison between results from VWSM and those from the method of characteristics (MC) is carried out. These results agree well with each other, and the applicability of VWSM into wave propagation study is verified. Subsequently, an interesting phenomenon, i.e., the magnitude of transmission coefficient increases with joint number on some circumstances, is pointed out and studied. By analyzing differently arriving transmitted waves and their superposition, such phenomenon is explained.

Validation study of the distinct lattice spring model (DLSM) on P-wave propagation across multiple parallel joints

A validation study of the distinct lattice spring model (DLSM) for wave propagation problems is performed. DLSM is a microstructure-based numerical model, which is meshless and has advantages in modelling dynamic problems where stress wave propagation is important. To verify the applicability of DLSM to modelling wave propagation through a discontinuous medium, the virtual wave source (VWS) method is used to obtain analytical solutions for wave propagation across a jointed rock mass. Numerical modelling results of the commercial code UDEC are selected as the reference. The effects of particle size and lattice rotation angle on wave propagation are first studied. Then, the results of wave transmission across a single joint with a different joint stiffness and across multiple parallel joints with different joint spacings are derived with DLSM, UDEC and VWS. These results are in good agreement with each other. Therefore, the capability of DLSM to model P-wave propagation across jointed rock mass is verified, which provides confidence for the further application of DLSM to modelling more complex problems.

Theoretical Methods for Wave Propagation across Jointed Rock Masses

Different methods are presently available for the analysis of wave propagation across jointed rock masses with the consideration of multiple wave reflections between joints. These methods can be divided into two categories. One is based on the displacement discontinuity model for representing rock joints, where the displacements across a joint are discontinuous and the tractions are continuous, and the other is the equivalent medium method. For the first category, there are three methods, i.e., method of characteristics (MC), scattering matrix method (SMM) and virtual wave source method (VWS). MC solves the equation of motion by using the theory of characteristic curves. SMM is based on the definition of the scattering matrix in which the reflection and transmission coefficients of a set of joints are stored. VWS method replaces the joints in the rock mass with a virtual concept. For the second category, equivalent medium model treats the problem in the frame of continuum mechanics and simplifies it from an explicit wave propagation equation. The objective of this paper is to review and compare these theoretical methods. The comparison shows that the four solutions agree very well with each other. Some additional considerations about the advantages and disadvantages of these methods are also given in the paper.

Virtual source method applied to crosswell and horizontal well geometries

The virtual source method is a useful tool for redatuming a seismic survey below complicated overburden by creating virtual sources at downhole receiver locations, hence generating data independent of the overburden and the time-lapse changes therein. In this article we first apply this technique to crosswell geometries, whereby a virtual crosswell survey is simulated by shooting a line of surface shots passing through two boreholes instrumented with downhole sensors. Using this acquisition geometry, receivers in one of the two wells are turned into virtual sources by correlating the wavefield recorded by those receivers with the recording at receivers in the other well, and summing the correlated data over the surface shots.

Analysis of One-dimensional Heat Diffusion after Light Pulse Heating by the Response Function Method

To measure thermal diffusivity across thin films, the laser flash method has been improved by using ultrafast lasers and thermoreflectance temperature detection. These light pulse heating methods allow observation of one-dimensional heat diffusion from the heated face to the detected face of a specimen across a well-defined length of specimen thickness following the light pulse heating. Since most electronic devices, storage media, displays, and so forth have a multilayered structure of thin films, information about heat diffusion across multilayers, thermophysical properties of constituent single layers, boundary thermal resistance between the layers, and effective thermophysical properties of the multilayered structure as a whole are required for improved thermal design of devices and development of materials. To meet these requirements, a generally applicable approach to analyze one-dimensional heat diffusion across multilayers has been developed on the basis of the virtual heat source method and response function method. These methods of analysis can be applied over a timescale ranging from picoseconds, as observed by the picosecond thermoreflectance method, to more than 10 seconds, as observed by the laser flash method.

Concept and Spatial Analysis Method of Urban Environmental Traffic Capacity

A method to determine the intensity of traffic is a key issue for the control of air pollution from urban transport in order to meet air quality requirements. In this study, by combining the definitions of environmental capacity and traffic capacity, the concepts of traffic-related environmental capacity and environmental traffic capacity are discussed. A mesh analysis method for environmental traffic capacity in urban areas has been developed by integrating the spatial segmentation method applied for the analysis of urban traffic system with the environmental spatial segmentation method. To calculate the spatial transfer coefficients of air pollutants between grids, a method of backing virtual point sources was established and applied to this case. The linear optimization model of traffic-related environmental capacity was established on a basis of the diffusion and transmission theories for air pollutants. Taking peak hours as an analytical period, we have established an urban environmental traffic capacity analysis model aimed at controlling various air pollutants emitted by vehicles. A case study shows that the model can analyze and calculate vehicle traffic volumes, which satisfy air quality requirements within specified temporal and spatial ranges.

Estimating and correcting the amplitude radiation pattern of a virtual source

In the virtual source (VS) method we crosscorrelate seismic recordings at two receivers to create a new data set as if one of these receivers were a virtual source and the other a receiver. We focus on the amplitudes and kinematics of VS data, generated by an array of active sources at the surface and recorded by an array of receivers in a borehole. The quality of the VS data depends on the radiation pattern of the virtual source, which in turn is controlled by the spatial aperture of the surface source distribution. Theory suggests that when the receivers are surrounded by multi-component sources completely filling a closed surface, then the virtual source has an isotropic radiation pattern and VS data possess true amplitudes. In practical applications, limited sourceaperture and deployment of a single source type create an anisotropic radiation pattern of the virtual source, leading to distorted amplitudes. This pattern can be estimated by autocorrelating the spatial Fourier transform of the downgoing wavefield in the special case of a laterally invariant medium. The VS data can be improved by deconvolving the VS data with the estimated amplitude radiation pattern in the frequency-wavenumber domain. This operation alters the amplitude spectrum but not the phase of the data. We can also steer the virtual source by assigning it a new desired amplitude radiation pattern, given sufficient illumination exists in the desired directions. Alternatively, time-gating the downgoing wavefield before crosscorrelation, already common practice in implementing the VS method, can improve the radiation characteristics of a virtual source.

Estimating interval shear-wave splitting from multicomponent virtual shear check shots

Measuring shear-wave splitting from vertical seismic profiling (VSP) data can benefit fracture and stress characterization as well as seismic processing and interpretation. The classic approach to measuring azimuthal anisotropy at depth involves layer stripping. Its inherent weakness is the need to measure and undo overburden effects before arriving at an anisotropy estimate at depth. That task is challenging when the overburden is complex and varies quickly with depth. Moreover, VSP receivers are rarely present all the way from the surface to the target. That necessitates the use of simplistic assumptions about the uninstrumented part of the overburden that limit the quality of the result. We propose a new technique for measuring shear-wave splitting at depth that does not require any knowledge of the overburden. It is based on a multicomponent version of the virtual source method in which each two-component (2-C) VSP receiver is turned into a 2-C shear source and recorded at deeper geophones. The resulting virtual data set is affected only by the properties of the medium between the receivers. A simple Alford rotation transforms the data set into fast and slow shear virtual check shots from which shear-wave splitting can be measured easily and accurately under arbitrarily complex overburden.

Transient interaction of a plane acoustic wave with an elastic orthotropic cylindrical shell

A reflected-afterflow virtual-source (RAVS) method is introduced to obtain the transient dynamic response of an orthotropic cylindrical shell with infinite length subjected to a step plane wave. The Sanders thin shell theory is used to define the shell's dynamic behavior and the linear acoustic plane wave assumption is adopted. The elastic transient and vibratory responses of the orthotropic cylindrical shells are intensively investigated, and some interesting features of the effects of the shell radius and thickness upon the nondimensional radial velocity, mid-surface strain, 0th mode radial displacement and 1st mode radial velocity are revealed.

Exact description of a cylindrically symmetrical complex-argument Laguerre-Gauss beam

Based on the perturbative series representation of a complex-source-point spherical wave an expression for cylindrically symmetrical complex-argument Laguerre-Gauss beams of radial order n is derived. This description acquires the accuracy up to any order of diffraction angle, and its first three corrected terms are in accordance with those given by Seshadri [Opt. Lett.27, 1872 (2002)] based on the virtual source method. Numerical results show that on the beam axis the number of orders of nonvanishing nonparaxial corrections is equal to n. Meanwhile a higher radial mode number n leads to a smaller convergent domain of radius.

Strengthening the virtual-source method for time-lapse monitoring

Time-lapse monitoring is a powerful tool for tracking subsurface changes resulting from fluid migration. Conventional time-lapse monitoring can be done by observing differences between two seismic surveys over the surveillance period. Along with the changes in the subsurface, differences in the two seismic surveys are also caused by variations in the near-surface overburden and acquisition discrepancies. The virtual-source method monitors below the time-varying near-surface by redatuming the data down to the subsurface receiver locations. It crosscorrelates the signal that results from surface shooting recorded by subsurface receivers placed below the near-surface. For the Mars field data, redatuming the recorded response down to the permanently placed ocean-bottom cable (OBC) receivers using the virtual-source method allows one to reconstruct a survey as if virtualsources were buried at the OBC receiver locations and the medium above them were a homogeneous half-space. Separating the recorded wavefields into upgoing and downgoing (up-down) waves before crosscorrelation makes the resultant virtual-source data independent of the time-varying near-surface (seawater). For time-lapse monitoring, varying source signature for the two surveys and for each shot is also undesirable. Deconvolving the prestack crosscorrelated data (correlation gather) by the power spectrum of the source-time function results in virtual-source data independent of the source signature. Incorporating up-down wavefield separation and deconvolution of the correlation gather by the source power spectrum into the virtual-source method suppresses the causes of nonrepeatability in the seawater along with acquisition and source signature discrepancies. This processing combination strengthens the virtual-source method for time-lapse monitoring.

Acquisition geometry requirements for generating virtual-source data

Using model and field data, this article reviews the virtual-source method and its acquisition geometry requirements. Before we go into the details of the acquisition geometry requirements, let us briefly review the basic concept and the advantages of the virtual-source method. A typical surface seismic experiment has sources on the surface to excite waves that propagate through the subsurface. Surface receivers record the reflected waves. In order to image the subsurface, we migrate the reflected wavefield recorded by the receivers, using an estimate of the subsurface velocity model. However, the near surface is usually complex, and the velocity is difficult to estimate. These near-surface inhomogeneities, if not represented in the migration velocity model, defocus the deeper image. In order to avoid the estimation of the near-surface velocity model, Bakulin and Calvert (2006) proposed the virtual-source method, a technique that uses cross-correlation of the wavefield recorded by a given pair of receiver...

Estimates of scattering strength for buried cylindrical targets ensonified by evanescent waves

It is known that low-frequency subcritical sound waves can significantly scatter from targets buried in a seabed due to the significant penetration depth of the incident evanescent wave. Past computational work on scattering by buried spherical shells has been done, for example, using a T-matrix Method [R. Lim et al., J. Acoust. Soc. Am. 93, 1762-1783 (1993)], a Virtual Source Method [I. Lucifredi and H. Schmidt., J. Acoust. Soc. Am. 120, 3566-3583 (2006)], or finite-element methods [Zampolli et al., J. Acoust. Soc. Am., in press]. In addition to high-fidelity results which are expected from the preceding numerical methods, it desirable to have approximate analytical/asymptotic predictions of multistatic scattering strength for a variety of homogeneous or layered buried targets. Focusing on buried cylinders of infinite or finite-length, we first compute scattering using an approximate method that makes use of separation of variables and neglects multiple scattering between the interface and the target. Results are compared with those generated using the Axiscat/NURC/COMSOL finite-element method. Asymptotic estimates are then presented for scattering strength for objects completely buried in the seafloor for a flat interface (Work sponsored by ONR and NURC.)

The Virtual Source Method – Verifying the concept using numerical and physical modelling

The Virtual Source Method (VSM) has been proposed for removing the effects caused by heterogeneities in the near surface. The method involves acquisition geometries similar to that of VSP, with receivers below the most troublesome part of the overburden. The time reversal principal is utilised to focus down-going energy through the overburden into useful primary energy at the virtual source location. The time reversal process is performed during data processing, and requires no knowledge of the velocity model between sources and sub-surface receivers. The result is a downward continued-dataset with virtual sources at the sub-surface receiver locations. The concept is verified using numerical and physical modelling, which demonstrates that the method can be used to accurately detect reflectors at depth, where conventional seismic fails.

Model-Based Correction of Diffraction Effects of the Virtual Source Element

A method for ultrasonic synthetic aperture imaging using finite-sized transducers is introduced that is based on a virtual source (VS) concept. In this setup, a focused transducer creates a VS element at its focal point that facilitates the use of synthetic aperture focusing technique (SAFT). It is shown that the performance of the VS method may be unsatisfactory due to the distortion introduced by the diffraction effects of the aperture used for creating the VS element. A solution to this problem is proposed that consists of replacing the classical SAFT by the extended synthetic aperature focusing technique (ESAFT) algorithm presented in our earlier works. In ESAFT, the full geometry of the VS is modeled, instead of applying the simplified point source approximation used when VS is combined with classical SAFT. The proposed method yields a substantial improvement in spatial resolution compared to that obtained using SAFT. Performance of the proposed algorithm is first demonstrated on simulated data, then verified on real data acquired with an array system.

Improving the virtual source method by wavefield separation

The virtual source method has recently been proposed to image and monitor below complex and time-varying overburden. The method requires surface shooting recorded at downhole receivers placed below the distorting or changing part of the overburden. Redatuming with the measured Green’s function allows the reconstruction of a complete downhole survey as if the sources were also buried at the receiver locations. There are still some challenges that need to be addressed in the virtual source method, such as limited acquisition aperture and energy coming from the overburden. We demonstrate that up-down wavefield separation can substantially improve the quality of virtual source data. First, it allows us to eliminate artifacts associated with the limited acquisition aperture typically used in practice. Second, it allows us to reconstruct a new optimized response in the absence of downgoing reflections and multiples from the overburden. These improvements are illustrated on a synthetic data set of a complex layered model modeled after the Fahud field in Oman, and on ocean-bottom seismic data acquired in the Mars field in the deepwater Gulf of Mexico.