Virtual Receiver Research Articles

Green's Function Retrieval and Marchenko Imaging in a Dissipative Acoustic Medium.

Single-sided Marchenko equations for Green's function construction and imaging relate the measured reflection response of a lossless heterogeneous medium to an acoustic wave field inside this medium. I derive two sets of single-sided Marchenko equations for the same purpose, each in a heterogeneous medium, with one medium being dissipative and the other a corresponding medium with negative dissipation. Double-sided scattering data of the dissipative medium are required as input to compute the surface reflection response in the corresponding medium with negative dissipation. I show that each set of single-sided Marchenko equations leads to Green's functions with a virtual receiver inside the medium: one exists inside the dissipative medium and one in the medium with negative dissipation. This forms the basis of imaging inside a dissipative heterogeneous medium. I relate the Green's functions to the reflection response inside each medium, from which the image can be constructed. I illustrate the method with a one-dimensional example that shows the image quality. The method has a potentially wide range of imaging applications where the material under test is accessible from two sides.

On the Optimization of a PSP-Based CPM Detection

This paper deals with a reduced-complexity per survivor processing-based CPM demodulation. It relies on a trellis with reduced state number and defined from a rational modulation index possibly different from the transmit modulation index and referred to as virtual receiver modulation index. The virtual receiver modulation index should be chosen so as to achieve a tradeoff between error-rate performance and complexity reduction. The main purpose of this paper is the choice of the virtual receiver modulation index. It gives guidelines to discard the values of the virtual receiver modulation index, that degrade the error-rate performance. Two criteria are used. The first one considers the uncoded CPM case and is based on an approximation of the minimal Euclidean distance. The second one is related to the bit interleaved coded CPM case and resorts to an EXIT chart to analyze the convergence of the iterative receiver.

Reconstructing the primary reflections in seismic data by Marchenko redatuming and convolutional interferometry

State-of-the-art methods to image the earth’s subsurface using active-source seismic reflection data involve reverse time migration. This and other standard seismic processing methods such as velocity analysis provide best results only when all waves in the data set are primaries (waves reflected only once). A variety of methods are therefore deployed as processing to predict and remove multiples (waves reflected several times); however, accurate removal of those predicted multiples from the recorded data using adaptive subtraction techniques proves challenging, even in cases in which they can be predicted with reasonable accuracy. We present a new, alternative strategy to construct a parallel data set consisting only of primaries, which is calculated directly from recorded data. This obviates the need for multiple prediction and removal methods. Primaries are constructed by using convolutional interferometry to combine the first-arriving events of upgoing and direct-wave downgoing Green’s functions to virtual receivers in the subsurface. The required upgoing wavefields to virtual receivers are constructed by Marchenko redatuming. Crucially, this is possible without detailed models of the earth’s subsurface reflectivity structure: Similar to the most migration techniques, the method only requires surface reflection data and estimates of direct (nonreflected) arrivals between the virtual subsurface sources and the acquisition surface. We evaluate the method on a stratified synclinal model. It is shown to be particularly robust against errors in the reference velocity model used and to improve the migrated images substantially.

Redatuming controlled-source electromagnetic data using Stratton–Chu type integral transformations

Abstract We present a new method of analyzing controlled-source electromagnetic (CSEM) data based on redatuming of the observed data from the actual receivers into the virtual receivers. We use the Stratton–Chu type integral transform to calculate the EM field in the virtual receivers. The virtual receivers can be placed at any desirable position, including close to the target, which increases the sensitivity of the EM data to the target. The developed method provides an effective model-based interpolation/extrapolation tool for electromagnetic field data. This paper demonstrates that redatuming can be used for designing the optimized CSEM survey configuration. The numerical examples, for the Kevin Dome Electromagnetic Project Site, illustrate the practical effectiveness of the developed method.

Retrieving virtual reflection responses at drill-bit positions using seismic interferometry with drill-bit noise

In the field of seismic interferometry, researchers have retrieved surface waves and body waves by cross-correlating recordings of uncorrelated noise sources to extract useful subsurface information. The retrieved wavefields in most applications are between receivers. When the positions of the noise sources are known, inter-source interferometry can be applied to retrieve the wavefields between sources, thus turning sources into virtual receivers. Previous applications of this form of interferometry assume impulsive point sources or transient sources with similar signatures. We investigate the requirements of applying inter-source seismic interferometry using non-transient noise sources with known positions to retrieve reflection responses at those positions and show the results using synthetic drilling noise as source. We show that, if pilot signals (estimates of the drill-bit signals) are not available, it is required that the drill-bit signals are the same and that the phases of the virtual reflections at drill-bit positions can be retrieved by deconvolution interferometry or by cross-coherence interferometry. Further, for this case, classic interferometry by cross-correlation can be used if the source power spectrum can be estimated. If pilot signals are available, virtual reflection responses can be obtained by first using standard seismic-while-drilling processing techniques such as pilot cross-correlation and pilot deconvolution to remove the drill-bit signatures in the data and then applying cross-correlation interferometry. Therefore, provided that pilot signals are reliable, drill-bit data can be redatumed from surface to borehole depths using this inter-source interferometry approach without any velocity information of the medium, and we show that a well-positioned image below the borehole can be obtained using interferometrically redatumed reflection responses with just a simple velocity model. We discuss some of the practical hurdles that restrict the application of the proposed method offshore.

A projection-based approach to diffraction tomography on curved boundaries

An approach to diffraction tomography is investigated for two-dimensional image reconstruction of objects surrounded by an arbitrarily-shaped curve of sources and receivers. Based on the integral theorem of Helmholtz and Kirchhoff, the approach relies upon a valid choice of the Green’s functions for selected conditions along the (possibly-irregular) boundary. This allows field projections from the receivers to an arbitrary external location. When performed over all source locations, it will be shown that the field caused by a hypothetical source at this external location is also known along the boundary. This field can then be projected to new external points that may serve as a virtual receiver. Under such a reformation, data may be put in a form suitable for image construction by synthetic aperture methods. Foundations of the approach are shown, followed by a mapping technique optimized for the approach. Examples formed from synthetic data are provided.

Receiver-pair seismic interferometry applied to body-wave USArray data

With seismic interferometry, reflections can be retrieved between stations positioned on the Earth's surface. In the classical form, the reflections are retrieved by a crosscorrelation of observations and an integration over subsurface sources. For a specific data set, however, the actual source distribution might not be sufficient to approximate the source integral. Yet, there might be a dense distribution of receivers allowing an integration over the receiver domain.We rewrite the source integral to an integration over receiver pairs and call it receiver-pair seismic interferometry (RPSI). With this formulation, reflections can be retrieved even in the limiting case of only a single source. We illustrate the new relation both with synthetic data and data from the USArray, which is a large grid of stations covering the USA. The field observations are from an earthquake in Mexico. We show that RPSI can be applied both for a line and grid of receivers. When using isolated phases recorded over a line of stations inline with the earthquake, reflections are retrieved from the core-mantle boundary, which reflections can be ascribed to specific virtual source and receiver locations within the USArray. When using full responses as input to RPSI, the retrieved phases are an average over multiple virtual sources and receivers. Location is then only possible when the integrand is spatially windowed or when a clear leading term is identified. When using a grid of receivers, the location of the source does not need to be known, but spatially averaged instead of localized responses are obtained, also when isolated arrivals are used as input to RPSI.

Marchenko imaging

Traditionally, the Marchenko equation forms a basis for 1D inverse scattering problems. A 3D extension of the Marchenko equation enables the retrieval of the Green’s response to a virtual source in the subsurface from reflection measurements at the earth’s surface. This constitutes an important step beyond seismic interferometry. Whereas seismic interferometry requires a receiver at the position of the virtual source, for the Marchenko scheme it suffices to have sources and receivers at the surface only. The underlying assumptions are that the medium is lossless and that an estimate of the direct arrivals of the Green’s function is available. The Green’s function retrieved with the 3D Marchenko scheme contains accurate internal multiples of the inhomogeneous subsurface. Using source-receiver reciprocity, the retrieved Green’s function can be interpreted as the response to sources at the surface, observed by a virtual receiver in the subsurface. By decomposing the 3D Marchenko equation, the response at the virtual receiver can be decomposed into a downgoing field and an upgoing field. By deconvolving the retrieved upgoing field with the downgoing field, a reflection response is obtained, with virtual sources and virtual receivers in the subsurface. This redatumed reflection response is free of spurious events related to internal multiples in the overburden. The redatumed reflection response forms the basis for obtaining an image of a target zone. An important feature is that spurious reflections in the target zone are suppressed, without the need to resolve first the reflection properties of the overburden.

Data-driven wavefield focusing and imaging with multidimensional deconvolution: Numerical examples for reflection data with internal multiples

Standard imaging techniques rely on the single scattering assumption. This requires that the recorded data do not include internal multiples, i.e., waves that have bounced multiple times between reflectors before reaching the receivers at the acquisition surface. When multiple reflections are present in the data, standard imaging algorithms incorrectly image them as ghost reflectors. These artifacts can mislead interpreters in locating potential hydrocarbon reservoirs. Recently, we introduced a new approach for retrieving the Green’s function recorded at the acquisition surface due to a virtual source located at depth. We refer to this approach as data-driven wavefield focusing. Additionally, after applying source-receiver reciprocity, this approach allowed us to decompose the Green’s function at a virtual receiver at depth in its downgoing and upgoing components. These wavefields were then used to create a ghost-free image of the medium with either crosscorrelation or multidimensional deconvolution, presenting an advantage over standard prestack migration. We tested the robustness of our approach when an erroneous background velocity model is used to estimate the first-arriving waves, which are a required input for the data-driven wavefield focusing process. We tested the new method with a numerical example based on a modification of the Amoco model.

Field and synthetic experiments for virtual source crosswell tomography in vertical wells: Perth Basin, Western Australia

It is common for at least one monitoring well to be located proximally to a production well. This presents the possibility of applying crosswell technologies to resolve a range of earth properties between the wells. We present both field and synthetic examples of dual well walk-away vertical seismic profiling in vertical wells and show how the direct arrivals from a virtual source may be used to create velocity images between the wells. The synthetic experiments highlight the potential of virtual source crosswell tomography where large numbers of closely spaced receivers can be deployed in multiple wells. The field experiment is completed in two monitoring wells at an aquifer storage and recovery site near Perth, Western Australia. For this site, the crosswell velocity distribution recovered from inversion of travel times between in-hole virtual sources and receivers is highly consistent with what is expected from sonic logging and detailed zero-offset vertical seismic profiling. When compared to conventional walkaway vertical seismic profiling, the only additional effort required to complete dual-well walkaway vertical seismic profiling is the deployment of seismic sensors in the second well. The significant advantage of virtual source crosswell tomography is realised where strong near surface heterogeneity results in large travel time statics. • VS crosswell method used for developing a hydrostratigraphic model for ASR site. • VS crosswell is a cost-effective and low-impact method. • VS crosswell methods help to characterise aquifers within the ASR injection zone.

The Improvement of 3D Traveltime Tomographic Inversion Method

As 3D high-precision seismic exploration is more and more widely used in seismic data acquisition, traveltime tomographic inversion based on first arrivals is developed from 2D to 3D. However, magnanimity data of 3D traveltime inversion brings about the problem of data storage; the absence of first arrivals with near offset reduces the precision of shallow layer; the utilization of prior information, such as small refraction and micro-logging data, can improve the precision of 3D traveltime inversion. Therefore, we make some improvements in 3D traveltime inversion method. We take compression storage for large and sparse matrix, propose virtual receivers technology, and add prior information to tomographic inversion linear equations. The application in 3D real data indicates that the improvements can effectively improve 3D traveltime tomographic inversion. Key words : 3D seismic exploration; 3D traveltime inversion method; 3D traveltime tomographic inversion

SCTP performance improvement based on virtual receiver window

Abstract This article puts forward one algorithm for stream control transfer protocol (SCTP) improvement with limited receiver buffer (RBUF). As is well known, SCTP is one of the most important transfer control protocol, but most researches focus on the situations without the RBUF limit. In this study, we analyze the impact of the RBUF size on the performance. Computer simulations show that the network utility is low in reliable transfer, when the RBUF size is smaller than bandwidth delay product. By studying the transmission sequence number (TSN) transfer progress, we find that the peer receiver window (PEER_RWND), which lags behind the true receiver window (RWND), leads to the poor network utility. To improve SCTP performance with limited RBUF, the virtual receiver window (VIRTUAL_RWND) is introduced. Based on the VIRTUAL_RWND, one algorithm is proposed to increase the sending rate. Computer simulations have evaluated an excellent performance of the proposed algorithm at both ideal link without lost packet and nonideal link with lost packet.

Interferometric imaging of the underside of a subducting crust

SUMMARY Seismic interferometry provides tools for redatuming physical data to a new source location. Turning a source, located close to a structure of interest, into a virtual receiver has the potential benefit of improving the quality of imaging by increasing the effective aperture and mitigatingtheeffectofvelocityuncertaintyintheoverburden.Here,weconsidertheproblemof estimatingtheGreen’sfunctionbetweentwoearthquakeslocatedinsideasubductingslabusing earthquake data recorded at the surface. Our primary focus is to obtain an accurate time-image of the subducting interface. We propose a novel two-step kinematically correct redatuming procedure that first redatums the data from earthquakes below the subducting interface to the surface via classical interferometry, and then utilizes source–receiver wavefield interferometry to redatum virtual surface seismic data to the location of a particular earthquake event.

On the sensitivity of surface NMR in the presence of electrical conductivity anomalies

The surface-NMR tomography technique is based on the principles of electromagnetic induction and proton spin dynamics. Electromagnetic fields emitted by large surface current-driven loops are employed to locate and quantify groundwater reservoirs. The oscillating magnetic fields interact with proton spins of water molecules in the electrically conductive subsurface. To study the influence of changing subsurface electrical properties on the nuclear spin response, we consider the spin magnetization as a virtual magnetic dipole receiver. The numerical solutions for the electric and magnetic fields of the transmitter and the virtual receiver in 3-D heterogeneous ground are based on the finite-element method. We explicitly compute the frequency-domain electromagnetic sensitivities for separate spin magnetizations in a groundwater aquifer to study the distortion of the NMR response because of electrical heterogeneities in the medium. Analyses of entire pulse moment sequences yield the cumulative sensitivities to electrical conductivity and water-content variations in the subsurface. We illustrate the influence of conductivity on NMR responses using a limited number of models. From these models we found that electrical conductivity anomalies in the shallow subsurface (<50 m) having values =0.1 S m-1 and volumes with linear dimensions in the order of our loop size (i.e. edge length 100 m) can have a strong influence on the NMR response and ought to be taken into account in the inversion of surface-NMR data. The effect increases non-linearly with increased body size, increased conductivity contrast and decreased anomaly depth.

Interferometric hydrofracture microseism localization using neighboring fracture

Hydraulic fracturing is the process of injecting high-pressure fluids into a reservoir to induce fractures and thus improve reservoir productivity. Microseismic event localization is used to locate created fractures. Traditionally, events are localized individually. Available information about events already localized is not used to help estimate other source locations. Traditional localization methods yield an uncertainty that is inversely proportional to the square root of the number of receivers. However, in applications where multiple fractures are created, multiple sources in a reference fracture may provide redundant information about unknown events in subsequent fractures that can boost the signal-to-noise ratio, improving estimates of the event positions. We used sources in fractures closer to the monitoring well to help localize events further away. It is known through seismic interferometry that with a 2D array of receivers, the traveltime between two sources may be recovered from a crosscorrelogram of two common source gathers. This allowed an event in the second fracture to be localized relative to an event in the reference fracture. A difficulty became evident when receivers are located in a single monitoring well. When the receiver array is 1D, classical interferometry cannot be directly employed because the problem becomes underdetermined. In our approach, interferometry was used to partially redatum microseismic events from the second fracture onto the reference fracture so that they can be used as virtual receivers, providing additional information complementary to that provided by the physical receivers. Our error analysis showed that, in addition to the gain obtained by having multiple physical receivers, the location uncertainty is inversely proportional to the square root of the number of sources in the reference fracture. Because the number of microseism sources is usually high, the proposed method will usually result in more accurate location estimates as compared with the traditional methods.

Source-receiver interferometry for seismic wavefield construction and ground-roll removal

Seismic interferometry describes the construction of unmeasured wavefield responses (or Green's functions) between two or more points by applying cross-correlation, deconvolution, or convolution to seismic data recordings. The practical implications are that, applying inter-receiver interferometry, a “virtual” (imaginary) source of energy can be created at the location to a real receiver by using energy recorded from surrounding sources. Similarly, by using inter-source interferometry, a virtual receiver can be created at the location of a real source by using energy recorded at surrounding receivers (Curtis et al., 2009). These two methods can be combined to create the new technique of source-receiver interferometry (Curtis, 2009; Curtis and Halliday, 2010; Halliday and Curtis, 2010) which synthesizes real-source to real-receiver Green's function estimates using only energy recorded at a surrounding boundary of receivers and from an additional surrounding boundary of sources. The boundary sources in each case can be active (such as an explosive or vibrating source) or passive (such as noise from anthropogenic activity or ocean waves). This paper describes the first real-data application of source-receiver interferometry, and demonstrates its potential to enhance existing methods of interferometric ground-roll suppression.

Introduction to this special section: Interferometry applications

If some ten years ago, one approached a highly skilled geophysicist randomly picked from our community and asked “How can you turn signals recorded at two receivers into what would be recorded if one acts as a source?”, or “How can we use random noise from unknown sources for imaging?”, or “How do we turn an earthquake into a buried virtual receiver?”, the answer most likely would lie somewhere between “Uh, I don't know,” to “What?! You can't do that!” However, in recent years these questions as well as others (possibly even stranger) have been answered due to a boom of creative scientific work in interferometry.

A Virtual Layered Space-Frequency Receiver Architecture with Iterative Decoding

In this paper, a novel receiver architecture is proposed for multiple-input-multiple-output (MIMO) systems; the proposed architecture helps achieve superior performance in multipath fading channels when the number of layered streams exceeds the number of receiving antennas. In this architecture, the concept of “virtual channel” is adopted to attain diversity gain even when successive detection is applied for reducing computational complexity, while orthogonal frequency division multiplexing (OFDM) is employed to combat multipath fading. Actually, successive detection is carried out in all possible virtual channels, and the virtual channel with the minimum error probability is detected with the assistance of the maximum a-posteriori (MAP) decoder in the architecture. In addition, soft input and soft output (SISO) iterative detection is introduced in the virtual channel estimation scheme. The performance of the proposed architecture is verified by computer simulations. This architecture can be implemented with lesser complexity than that in maximum likelihood detection (MLD), but the gain in the former case exceeds that in the latter by 4.5dB at the BER of 10-3 for 4×2 MIMO-OFDM.

Multiple-Input Multiple-Output Gaussian Broadcast Channels With Common and Confidential Messages

This paper considers the problem of the multiple-input multiple-output (MIMO) Gaussian broadcast channel with two receivers (receivers 1 and 2) and two messages: a common message intended for both receivers and a confidential message intended only for receiver 1 but needing to be kept asymptotically perfectly secure from receiver 2. A matrix characterization of the secrecy capacity region is established via a channel enhancement argument. The enhanced channel is constructed by first splitting receiver 1 into two virtual receivers and then enhancing only the virtual receiver that decodes the confidential message. The secrecy capacity region of the enhanced channel is characterized using an extremal entropy inequality previously established for characterizing the capacity region of a degraded compound MIMO Gaussian broadcast channel.

An optimal orthogonal overlay for a cyclostationary legacy signal

In this paper, the design of an overlay system is considered that operates on a channel already occupied by a legacy system. It is assumed that the legacy transmitter generates a cyclostationary signal, and that the legacy receiver is equipped with a linear filter followed by a uniform sampler. Under a zero-interference constraint at the legacy receiver, the meansquared error at the overlay receiver is minimized to find the optimal overlay system that employs linear modulation and demodulation. For the trade-off between performance and rate of the overlay system, the problem is formulated to allow a transmission rate that is a fraction of the fundamental cycle frequency of the legacy signal. The notion of a virtual legacy receiver is devised to incorporate cases with non-identical frequency bands for the transmission and reception of the overlay signal. Using the vectorized Fourier transform technique, the problem is reformulated, the solution is derived, and a necessary and sufficient condition for the existence of the optimal solution is found, all in the frequency domain. Numerical results show that the proposed scheme significantly improves the spectral efficiency by effectively exploiting the cyclostationarity of the legacy signal.