Equivalence of the virtual-source method and wave-field deconvolution in seismic interferometry

Investigation of Time-Lapse Changes with DAS Borehole Data at the Brady Geothermal Field Using Deconvolution Interferometry

Distributed acoustic sensing (DAS) has great potential for monitoring natural-resource reservoirs and borehole conditions. However, the large volume of data and complicated wavefield add challenges to processing and interpretation. In this study, we demonstrate that seismic interferometry based on deconvolution is a convenient tool for analyzing this complicated wavefield. We also show the limitation of this technique, in that it still requires good coupling to extract the signal of interest. We extract coherent waves from the observation of a borehole DAS system at the Brady geothermal field in Nevada. The extracted waves are cable or casing ringing that reverberate within a depth interval. These ringing phenomena are frequently observed in the vertical borehole DAS data. The deconvolution method allows us to examine the wavefield at different boundary conditions and separate the direct waves and the multiples. With these benefits, we can interpret the wavefields using a simple 1D string model and monitor its temporal changes. The velocity of this wave varies with depth, observation time, temperature, and pressure. We find the velocity is sensitive to disturbances in the borehole related to increasing operation intensity. The velocity decreases with rising temperature. The reverberation can be decomposed into distinct vibration modes in the spectrum. We find that the wave is dispersive and the fundamental mode propagates with a large velocity. This interferometry method can be useful for monitoring borehole conditions or reservoir property changes using densely-sampled DAS data.

Time-Domain Multidimensional Deconvolution: A Physically Reliable and Stable Preconditioned Implementation

Multidimensional deconvolution constitutes an essential operation in a variety of geophysical scenarios at different scales ranging from reservoir to crustal, as it appears in applications such as surface multiple elimination, target-oriented redatuming, and interferometric body-wave retrieval just to name a few. Depending on the use case, active, microseismic, or teleseismic signals are used to reconstruct the broadband response that would have been recorded between two observation points as if one were a virtual source. Reconstructing such a response relies on the the solution of an ill-conditioned linear inverse problem sensitive to noise and artifacts due to incomplete acquisition, limited sources, and band-limited data. Typically, this inversion is performed in the Fourier domain where the inverse problem is solved per frequency via direct or iterative solvers. While this inversion is in theory meant to remove spurious events from cross-correlation gathers and to correct amplitudes, difficulties arise in the estimation of optimal regularization parameters, which are worsened by the fact they must be estimated at each frequency independently. Here we show the benefits of formulating the problem in the time domain and introduce a number of physical constraints that naturally drive the inversion towards a reduced set of stable, meaningful solutions. By exploiting reciprocity, time causality, and frequency-wavenumber locality a set of preconditioners are included at minimal additional cost as a way to alleviate the dependency on an optimal damping parameter to stabilize the inversion. With an interferometric redatuming example, we demonstrate how our time domain implementation successfully reconstructs the overburden-free reflection response beneath a complex salt body from noise-contaminated up- and down-going transmission responses at the target level.

Reflecting boundary conditions for interferometry by multidimensional deconvolution

In an acoustical context, interferometry takes advantage of existing (ambient) wavefield recordings by turning receivers into so-called "virtual sources." The medium's response to these virtual sources can be harnessed to image that medium. Most interferometric applications, however, suffer from the fact that the retrieved virtual-source responses deviate from the true medium responses. The accrued artefacts are often predominantly due to a non-isotropic illumination of the medium of interest, and prohibit accurate interferometric imaging. Recently, it has been shown that illumination-related artefacts can be removed by means of a so-called multidimensional deconvolution (MDD) process. However, the current MDD formulation, and hence method, relies on separation of waves traveling inward and outward through the boundary of the medium of interest. As a consequence, it is predominantly useful when receivers are illuminated from one side only. This puts constraints on the applicability of the current MDD formulation to omnidirectional wavefields. In this paper, a modified formulation of the theory underlying interferometry by MDD is presented. This modified formulation eliminates the requirement to separate inward and outward propagating wavefields and, consequently, holds promise for the application of MDD to non-isotropic, omnidirectional wavefields.

Green's Function Retrieval with Absorbing Probes in Reverberating Cavities

The cross-correlation of a diffuse wave field converges toward the difference between the anticausal and causal Green's functions between two points. This property has paved the way to passive imaging using ambient noise sources. In this Letter, we investigate Green's function retrieval in electromagnetism. Using a model based on the fluctuation dissipation theorem, we demonstrate theoretically that the cross-correlation function strongly depends on the absorption properties of the receivers. This is confirmed in measurements within a reverberation chamber. In contrast to measurements with noninvasive probes, we show that only the anticausal Green's function can be retrieved with a matched antenna. Finally, we interpret this result as an equivalent time-reversal experiment with an electromagnetic sink.

Bridging Three Orders of Magnitude: Multiple Scattered Waves Sense Fractal Microscopic Structures via Dispersion

Wave scattering provides profound insight into the structure of matter. Typically, the ability to sense microstructure is determined by the ratio of scatterer size to probing wavelength. Here, we address the question of whether macroscopic waves can report back the presence and distribution of microscopic scatterers despite several orders of magnitude difference in scale between wavelength and scatterer size. In our analysis, monosized hard scatterers 5  μm in radius are immersed in lossless gelatin phantoms to investigate the effect of multiple reflections on the propagation of shear waves with millimeter wavelength. Steady-state monochromatic waves are imaged in situ via magnetic resonance imaging, enabling quantification of the phase velocity at a voxel size big enough to contain thousands of individual scatterers, but small enough to resolve the wavelength. We show in theory, experiments, and simulations that the resulting coherent superposition of multiple reflections gives rise to power-law dispersion at the macroscopic scale if the scatterer distribution exhibits apparent fractality over an effective length scale that is comparable to the probing wavelength. Since apparent fractality is naturally present in any random medium, microstructure can thereby leave its fingerprint on the macroscopically quantifiable power-law exponent. Our results are generic to wave phenomena and carry great potential for sensing microstructure that exhibits intrinsic fractality, such as, for instance, vasculature.

Lagrangian and energy forms for retrieving the impulse response of the Earth due to random electromagnetic forcing

We distinguish between trivial and nontrivial differences in retrieving the real or imaginary parts of the Green's function. Trivial differences come from different Green's function definitions. The energy and lagrangian forms constitute nontrivial differences. Magnetic noise sources suffice to extract the quasistatic electromagnetic-field Earth impulse response in the lagrangian form. This is of interest for Earth subsurface imaging. A numerical example demonstrates that all source vector components are necessary to extract a single-field vector component.

A representation for Green's function retrieval by multidimensional deconvolution

Green's function retrieval by crosscorrelation may suffer from irregularities in the source distribution, asymmetric illumination, intrinsic losses, etc. Multidimensional deconvolution (MDD) may overcome these limitations. A unified representation for Green's function retrieval by MDD is proposed. From this representation, it follows that the traditional crosscorrelation method gives a Green's function of which the source is smeared in space and time. This smearing is quantified by a space-time point-spread function (PSF), which can be retrieved from measurements at an array of receivers. MDD removes this PSF and thus deblurs and deghosts the source of the Green's function obtained by correlation.

Nondiffuse elastic and anelastic passive imaging

The property at the basis of passive acoustic imaging is that, taken any two points, one of them can be seen as the source of the waves and the other as the recording station. This property, which was shown to hold also in nondiffuse fields, is here exploited: (1) to allow an undistorted passive imaging through the simple use of the statistical mode to estimate wave velocity, (2) to determine the azimuth of the instantaneous Huygens sources of the noise wavefield, and (3) to measure, provided that the noise bandwidth is wide with respect to that of the local system, the material dissipation constant as a function of frequency. The authors applied this theory to study the seismic noise field in the Ravenna, North-Central Italy, shore area and found it capable to provide velocity dispersion curves matching those of independent surveys, to track the sources of seismic noise to a few major firms in Ravenna port, with the prevailing source switching at the time scale of seconds, and to measure the dissipation quality factor Q at approximately 20 independent of frequency in the range 1-30 Hz.

Virtual reflector representation theorem (acoustic medium)

The virtual reflector method simulates new seismic signals by processing traces recorded by a plurality of sources and receivers. The approach is based on the crossconvolution of the recorded signals and makes it possible to obtain the Green's function of virtual reflected signals as if in the position of the receivers (or sources) there were a reflector, even if said reflector is not present. This letter presents the virtual reflector theory based on the Kirchhoff integral representation theorem for wave propagation in an acoustic medium with and without boundary and a generalization to variable reflection coefficients for scattered wavefields.

Cancellation of spurious arrivals in Green's function extraction and the generalized optical theorem

The extraction of the Green's function by cross correlation of waves recorded at two receivers nowadays finds much application. We show that for an arbitrary small scatterer, the cross terms of scattered waves give an unphysical wave with an arrival time that is independent of the source position. This constitutes an apparent inconsistency because theory predicts that such spurious arrivals do not arise, after integration over a complete source aperture. This puzzling inconsistency can be resolved for an arbitrary scatterer by integrating the contribution of all sources in the stationary phase approximation to show that the stationary phase contributions to the source integral cancel the spurious arrival by virtue of the generalized optical theorem. This work constitutes an alternative derivation of this theorem. When the source aperture is incomplete, the spurious arrival is not canceled and could be misinterpreted to be part of the Green's function. We give an example of how spurious arrivals provide information about the medium complementary to that given by the direct and scattered waves; the spurious waves can thus potentially be used to better constrain the medium.

Extracting the Green's function of attenuating heterogeneous acoustic media from uncorrelated waves

The Green's function of acoustic or elastic wave propagation can, for loss-less media, be retrieved by correlating the wave field that is excited by random sources and is recorded at two locations. Here the generalization of this idea to attenuating acoustic waves in an inhomogeneous medium is addressed, and it is shown that the Green's function can be retrieved from waves that are excited throughout the volume by spatially uncorrelated injection sources with a power spectrum that is proportional to the local dissipation rate. For a finite volume, one needs both volume sources and sources at the bounding surface for the extraction of the Green's functions. For the special case of a homogeneous attenuating medium defined over a finite volume, the phase and geometrical spreading of the Green's function is correctly retrieved when the volume sources are ignored, but the attenuation is not.

Using cross correlations of turbulent flow-induced ambient vibrations to estimate the structural impulse response. Application to structural health monitoring

It has been demonstrated theoretically and experimentally that an estimate of the impulse response (or Green's function) between two receivers can be obtained from the cross correlation of diffuse wave fields at these two receivers in various environments and frequency ranges: ultrasonics, civil engineering, underwater acoustics, and seismology. This result provides a means for structural monitoring using ambient structure-borne noise only, without the use of active sources. This paper presents experimental results obtained from flow-induced random vibration data recorded by pairs of accelerometers mounted within a flat plate or hydrofoil in the test section of the U.S. Navy's William B. Morgan Large Cavitation Channel. The experiments were conducted at high Reynolds number (Re > 50 million) with the primary excitation source being turbulent boundary layer pressure fluctuations on the upper and lower surfaces of the plate or foil. Identical deterministic time signatures emerge from the noise cross-correlation function computed via robust and simple processing of noise measured on different days by a pair of passive sensors. These time signatures are used to determine and/or monitor the structural response of the test models from a few hundred to a few thousand Hertz.

Unified Green's function retrieval by cross-correlation; connection with energy principles

It has been shown theoretically and observationally that the Green's function for acoustic and elastic waves can be retrieved by cross-correlating fluctuations recorded at two locations. We extend the concept of the extraction of the Green's function to a wide class of scalar linear systems. For systems that are not invariant under time reversal, the fluctuations must be excited by volume sources in order to satisfy the energy balance (equipartitioning) that is needed to extract the Green's function. The general theory for retrieving the Green's function is illustrated with examples that include the diffusion equation, Schrödinger's equation, a vibrating string, the acoustic wave equation, a vibrating beam, and the advection equation. Examples are also shown of situations where the Green's function cannot be extracted from ambient fluctuations. The general theory opens up new applications for the extraction of the Green's function from field correlations that include flow in porous media, quantum mechanics, and the extraction of the response of mechanical structures such as bridges.

Retrieving the Green's function of the diffusion equation from the response to a random forcing

It is known that the Green's function for nondissipative acoustic or elastic wave propagation can be extracted by correlating noise recorded at different receivers. This property is often related to the invariance for time reversal of the acoustic or elastic wave equations. The diffusion equation is not invariant for time reversal. It is shown in this work that the Green's function of the diffusion equation can also be retrieved by correlating solutions of the diffusion equation that are excited randomly and are recorded at different locations. This property can be used to retrieve the Green's function for diffusive systems from ambient fluctuations. Potential applications include the fluid pressure in porous media, electromagnetic fields in conducting media, the diffusive transport of contaminants, and the intensity of multiply scattered waves.