Multidimensional Deconvolution Research Articles

Shallow virtual source redatuming by multi-dimensional deconvolution

Recently, new on-shore acquisition designs have been presented with multicomponent sensors deployed in the shallow sub-surface (20 m–60 m). Virtual source redatuming has been proposed for these data to compensate for surface statics and to enhance survey repeatability. In this paper, we investigate the feasibility of replacing the correlation-based formalism that undergirds virtual source redatuming with multi-dimensional deconvolution, offering various advantages such as the elimination of free-surface multiples and the potential to improve virtual source repeatability. To allow for data-driven calibration of the sensors and to improve robustness in cases with poor sensor spacing in the shallow sub-surface (resulting in a relatively high wavenumber content), we propose a new workflow for this configuration. We assume a dense source sampling and target signals that arrive at near-vertical propagation angles. First, the data are preconditioned by applying synthetic-aperture-source filters in the common receiver domain. Virtual source redatuming is carried out for the multi-component recordings individually, followed by an intermediate deconvolution step. After this specific pre-processing, we show that the downgoing and upgoing constituents of the wavefields can be separated without knowledge of the medium parameters, the source wavelet, or sensor characteristics. As a final step, free-surface multiples can be eliminated by multi-dimensional deconvolution of the upgoing fields with the downgoing fields.

Seismic interferometry by multidimensional deconvolution without wavefield separation

S U M M A R Y Seismic interferometry comprises a suite of methods to redatum recorded wavefields to those that would have been recorded if different sources (so-called virtual sources) had been activated. Seismic interferometry by cross-correlation has been formulated using either two-way (for full wavefields) or one-way (for directionally decomposed wavefields) representation theorems. To obtain improved Green’s function estimates, the cross-correlation result can be deconvolved by a quantity that identifies the smearing of the virtual source in space and time, the so-called point-spread function. This type of interferometry, known as interferometry by multidimensional deconvolution (MDD), has so far been applied only to one-way directionally decomposed fields, requiring accurate wavefield decomposition from dual (e.g. pressure and velocity) recordings. Here we propose a form of interferometry by multidimensional deconvolution that uses full wavefields with two-way representations, and simultaneously invert for pressure and (normal) velocity Green’s functions, rather than only velocity responses as for its one-way counterpart. Tests on synthetic data show that two-way MDD improves on results of interferometry by cross-correlation, and generally produces estimates of similar quality to those obtained by one-way MDD, suggesting that the preliminary decomposition into upand downgoing components of the pressure field is not required if pressure and velocity data are jointly used in the deconvolution. We also show that constraints on the directionality of the Green’s functions sought can be added directly into the MDD inversion process to further improve two-way multidimensional deconvolution. Finally, as a by-product of having pressure and particle velocity measurements, we adapt oneand two-way representation theorems to convert any particle velocity receiver into its corresponding virtual dipole/gradient source by means of MDD. Thus data recorded from standard monopolar (e.g. marine) pressure sources can be converted into data from dipolar (derivative) sources at no extra acquisition cost.

Retrieving surface waves from ambient seismic noise using seismic interferometry by multidimensional deconvolution

AbstractRetrieving virtual source surface waves from ambient seismic noise by cross correlation assumes, among others, that the noise field is equipartitioned and the medium is lossless. Violation of these assumptions reduces the accuracy of the retrieved waves. A point‐spread function computed from the same ambient noise quantifies the associated virtual source's spatial and temporal smearing. Multidimensional deconvolution (MDD) of the retrieved surface waves by this function has been shown to improve the virtual source's focusing and the accuracy of the retrieved waves using synthetic data. We tested MDD on data recorded during the Batholiths experiment, a passive deployment of broadband seismic sensors in British Columbia, Canada. The array consisted of two approximately linear station lines. Using 4 months of recordings, we retrieved fundamental‐mode Rayleigh waves (0.05–0.27 Hz). We only used noise time windows dominated by waves that traverse the northern line before reaching the southern (2.5% of all data). Compared to the conventional cross‐correlation result based on this subset, the MDD waveforms are better localized and have significantly higher signal‐to‐noise ratio. Furthermore, MDD corrects the phase, and the spatial deconvolution fills in a spectral (f, k domain) gap between the single‐frequency and double‐frequency microseism bands. Frequency whitening of the noise also fills the gap in the cross‐correlation result, but the signal‐to‐noise ratio of the MDD result remains higher. Comparison of the extracted phase velocities shows some differences between the methods, also when all data are included in the conventional cross correlation.

Point‐spread functions for interferometric imaging

ABSTRACTInterferometric redatuming is a data‐driven method to transform seismic responses with sources at one level and receivers at a deeper level into virtual reflection data with both sources and receivers at the deeper level. Although this method has traditionally been applied by cross‐correlation, accurate redatuming through a heterogeneous overburden requires solving a multidimensional deconvolution problem. Input data can be obtained either by direct observation (for instance in a horizontal borehole), by modelling or by a novel iterative scheme that is currently being developed. The output of interferometric redatuming can be used for imaging below the redatuming level, resulting in a so‐called interferometric image. Internal multiples from above the redatuming level are eliminated during this process. In the past, we introduced point‐spread functions for interferometric redatuming by cross‐correlation. These point‐spread functions quantify distortions in the redatumed data, caused by internal multiple reflections in the overburden. In this paper, we define point‐spread functions for interferometric imaging to quantify these distortions in the image domain. These point‐spread functions are similar to conventional resolution functions for seismic migration but they contain additional information on the internal multiples in the overburden and they are partly data‐driven. We show how these point‐spread functions can be visualized to diagnose image defocusing and artefacts. Finally, we illustrate how point‐spread functions can also be defined for interferometric imaging with passive noise sources in the subsurface or with simultaneous‐source acquisition at the surface.

Simulation of GPR passive interferometry using cross-correlation for LNAPL model monitoring application

Passive interferometry technology is based on the relation between the reflection and the transmission responses of the subsurface. The transmission response can be received at surface in the presence of the ambient noise source in the subsurface with the cross-correlation (CC) or multidimensional deconvolution methods. We investigate the feasibility of electromagnetic (EM) wave passive interferometry with CC method. We design a 2-D finite-difference time domain (FDTD) algorithm to simulate the long-duration ground penetrating radar (GPR) measurements with random distribution of passive EM sources. The noise sources have random duration time, waveform and spatial distribution. We test the FDTD GPR passive interferometry code with above source characteristics and apply the method to light non-aqueous phase liquid (LNAPL) monitoring. Based on the model simulation data, by using common midpoint velocity analysis and normal move out correction to process the interferometry retrieve record, we can accurately obtain the dynamic changing characteristics of the target's permittivity. The LNAPL dynamic leakage model can be imaged as well. The synthetic results demonstrate that the GPR passive interferometry is feasible in subsurface LNAPL monitoring. Our work provides a foundation for a passive interferometry field application using GPR.

Body-wave interferometry using regional earthquakes with multidimensional deconvolution after wavefield decomposition at free surface

Body-wave interferometry using regional earthquakes with multidimensional deconvolution after wavefield decomposition at free surface

Multidimensional deconvolution and processing of seismic-interferometry Arctic data

Multidimensional deconvolution (MDD) by point-spread-function removes the blurring effects and the spread distortions typically generated in the signal representation by seismic interferometry (SI). Under suitable conditions, the MDD inversion of SI signals reconstructed by the Kirchhoff-Helmholtz integral of crosscorrelation type is a valuable and robust technique to recover the Green’s function of the subsurface. A basic requirement for the effective MDD application to SI data is to know the separated incoming and outcoming wavefields at the receivers illuminated by the real sources. We extended the MDD concept to the virtual reflector (VR) approach by crossconvolution and compared equivalent results obtained with approximations for the propagated wavefields. Examples were discussed with Arctic data of an on-ice shallow-water seismic experiment affected by strong and dispersive flexural ice wave noise. The target was to improve the signal-to-noise ratio by redatuming the sources at the sea bottom. First, we processed the raw input signals recorded by sea-bottom receivers to obtain an approximation of the incoming wavefield from on-ice sources. Then, we processed the data of the whole 2D seismic line in two different ways: applying MDD by crosscorrelation and by crossconvolution to, backward, SI and, forward, VR interferometric results, respectively. The analysis of the signals redatumed to the sea bottom showed that the flexural ice noise was significantly mitigated with respect to the conventional interferometry approach, with improved reflections for seismic investigation purposes. The agreement of the phase in the SI and VR results after MDD inversion confirmed that the approach was robust and enabled us to enhance the signals combining the stacked sections obtained by SI MDD and VR MDD. The resulting MDD SI stacked sections showed an improvement in signal quality especially at low frequencies with respect to the stacked section obtained by conventional processing of the original data.

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.

Data‐driven Green's function retrieval and application to imaging with multidimensional deconvolution

An iterative method is presented that allows one to retrieve the Green's function originating from a virtual source located inside a medium using reflection data measured only at the acquisition surface. In addition to the reflection response, an estimate of the travel times corresponding to the direct arrivals is required. However, no detailed information about the heterogeneities in the medium is needed. The iterative scheme generalizes the Marchenko equation for inverse scattering to the seismic reflection problem. To give insight in the mechanism of the iterative method, its steps for a simple layered medium are analyzed using physical arguments based on the stationary phase method. The retrieved Green's wavefield is shown to correctly contain the multiples due to the inhomogeneities present in the medium. Additionally, a variant of the iterative scheme enables decomposition of the retrieved wavefield into its downgoing and upgoing components. These wavefields then enable creation of a ghost‐free image of the medium with either cross correlation or multidimensional deconvolution, presenting an advantage over standard prestack migration.

Surface wave retrieval in layered media using seismic interferometry by multidimensional deconvolution

Virtual-source surface wave responses can be retrieved using the crosscorrelation (CC) of wavefields observed at two receivers. Higher mode surface waves cannot be properly retrieved when there is a lack of subsurface sources that excite these wavefields, as is often the case. In this paper, we present a multidimensional-deconvolution (MDD) scheme that is based on an approximate convolution theorem. The scheme introduces an additional processing step in which the CC result is deconvolved by a so-called point-spread tensor. The involved point-spread functions capture the imprint of the lack of subsurface sources and possible anelastic effects, and quantify the associated spatial and temporal smearing of the virtualsource components that leads to the poor surfacewave retrieval. The functions can be calculated from the same wavefields as used in the CC method. For a 2-D example that is representative of the envisaged applications, we show that the deconvolution partially corrects for the smearing. The retrieved virtual-source response only has some amplitude error in the ideal situation of having the depth of the required vertical array equal to the depth penetration of the surface waves. The error is due to ignored cross-mode terms in the approximate convolution theorem. Shorter arrays are also possible. In the limit case of only a single surface receiver, the retrieved virtual-source response is still more accurate than the CC result. The MDD scheme is valid for horizontally layered media that are laterally invariant, and includes exclusively multicomponent point-force responses (rather than their spatial derivatives) and multicomponent observations. The improved retrieval of multimode surface waves can facilitate dispersion analyses in shallow-subsurface inversion problems and monitoring, and surface wave removal algorithms.

Singular-value decomposition analysis of source illumination in seismic interferometry by multidimensional deconvolution

We have developed a method to analytically evaluate the relationship between the source-receiver configuration and the retrieved wavefield in seismic interferometry performed by multidimensional deconvolution (MDD). The MDD method retrieves the wavefield with the desired source-receiver configuration from the observed wavefield without source information. We used a singular-value decomposition (SVD) approach to solve the inverse problem of MDD. By introducing SVD into MDD, we obtained quantities that revealed the characteristics of the MDD inverse problem and interpreted the effect of the initial source-receiver configuration for a survey design. We numerically simulated the wavefield with a 2D model and investigated the rank of the incident field matrix of the MDD inverse problem. With a source array of identical length, a sparse and a dense source distribution resulted in an incident field matrix of the same rank and retrieved the same wavefield. Therefore, the optimum source distribution can be determined by analyzing the rank of the incident field matrix of the inverse problem. In addition, the introduction of scatterers into the model improved the source illumination and effectively increased the rank, enabling MDD to retrieve a better wavefield. We found that the ambiguity of the wavefield inferred from the model resolution matrix was a good measure of the amount of illumination of each receiver by the sources. We used the field data recorded at the two boreholes from the surface sources to support our results of the numerical modeling. We evaluated the rank of incident field matrix with the dense and sparse source distribution. We discovered that these two distributions resulted in an incident field matrix of almost the same rank and retrieved almost the same wavefield as the numerical modeling. This is crucial information for designing seismic experiments using the MDD-based approach.

Electromagnetic interferometry in wavenumber and space domains in a layered earth

With interferometry applied to controlled-source electromagnetic data, the direct field and the airwave and all other effects related to the air-water interface can be suppressed in a data-driven way. Interferometry allows for retreival of the scattered field Green’s function of the subsurface or, in other words, the subsurface reflection response. This reflection response can then be further used to invert for the subsurface conductivity distribution. To perform interferometry in 3D, measurements on an areal grid are necessary. We discuss 3D interferometry by multidimensional deconvolution in the frequency-wavenumber and in the frequency-space domains and provide examples for a layered earth model. We use the synthetic aperture source concept to damp the signal at high wavenumbers to allow large receiver sampling distances. Interferometry indeed increases the detectability of a subsurface reservoir. Finally, we discuss the dependency of the accuracy of the retrieved reflection response on the two crucial parameters: the conductivity of the seabed at the receiver location and the stabilization parameter of the least-squares inversion.

Broadband extended images by joint inversion of multiple blended wavefields

Estimation of seismic velocities and subsurface reservoir properties in deep, complex, geologic environments calls for the coordination of new acquisition technology with novel depth-domain inversion methods. Here, we propose a method for inversion of subsurface reflectivity image gathers that jointly relies on broadband data acquired by multimeasurement methods (vector-acoustic data; pressure and its gradients) from monopole (pressure) and dipole (e.g., gradient) sources. Our inversion retrieves depth-domain extended images (EIs), which represent the full nonlinear reflectivity operators with pseudosources and pseudoreceivers within the subsurface, as a function of time or frequency. Based on recent advances in interferometry by multidimensional deconvolution (MDD), we present two MDD-based imaging conditions for an EI inversion. One imaging condition consists of deconvolving correlation-based EIs with the so-called joint point-spread function (JPSF). These methods can, in principle, account for imaging primaries as well as internal and free-surface multiples. Because it is based on MDD, our JPSF approach can fully account for blended/simultaneous-source data in imaging with no need to deblend/separate the simultaneous-source data prior to imaging. Using dual-source vector-acoustic data, we describe how the EI JPSF system is constructed by separating source and receiver wavefields from receiver-side upgoing and ghost data, from pressure and gradient sources. With numerical examples, we demonstrate how the method successfully inverts for EIs representing subsurface reflectivity, while benefiting from the increase in temporal and spatial bandwidth brought on by the joint use of dual-source vector-acoustic data. Although here we focus on broadband streamer seismic applications, our joint wavefield inversion approach provides a framework for jointly imaging data from multiple experiments of any kind (e.g., surface and borehole, active and passive) with acoustic, elastic, and electromagnetic fields.

Interferometric redatuming by sparse inversion

Abstract Assuming that transmission responses are known between the surface and a particular depth level in the subsurface, seismic sources can be effectively mapped to this level by a process called interferometric redatuming. After redatuming, the obtained wavefields can be used for imaging below this particular depth level. Interferometric redatuming consists of two steps, namely (i) the decomposition of the observed wavefields into downgoing and upgoing constituents and (ii) a multidimensional deconvolution of the upgoing constituents with the downgoing constituents. While this method works in theory, sensitivity to noise and artefacts due to incomplete acquisition require a different formulation. In this letter, we demonstrate the benefits of formulating the two steps that undergird interferometric redatuming in terms of a transform-domain sparsity-promoting program. By exploiting compressibility of seismic wavefields in the curvelet domain, the method not only becomes robust with respect to noise but we are also able to remove certain artefacts while preserving the frequency content. Although we observe improvements when we promote sparsity in the redatumed data space, we expect better results when interferometric redatuming would be combined or integrated with least-squares migration with sparsity promotion in the image space.

Data matching for free-surface multiple attenuation by multidimensional deconvolution

A common strategy for surface-related multiple elimination of seismic data is to predict multiples by a convolutional model and subtract these adaptively from the input gathers. Problems can be posed by interfering multiples and primaries. Removing multiples by multidimensional deconvolution (MDD) (inversion) does not suffer from these problems. However, this approach requires data to be consistent, which is often not the case, especially not at interpolated near-offsets. A novel method is proposed to improve data consistency prior to inversion. This is done by backpropagating first-order multiples with a time-gated reference primary event and matching these with early primaries in the input gather. After data matching, multiple elimination by MDD can be applied with a deterministic inversion scheme.

On the relation between seismic interferometry and the simultaneous‐source method

ABSTRACTIn seismic interferometry the response to a virtual source is created from responses to sequential transient or simultaneous noise sources. Most methods use crosscorrelation, but recently seismic interferometry by multidimensional deconvolution (MDD) has been proposed as well. In the simultaneous‐source method (also known as blended acquisition), overlapping responses to sources with small time delays are recorded. The crosstalk that occurs in imaging of simultaneous‐source data can be reduced by using phase‐encoded sources or simultaneous noise sources, by randomizing the time interval between the shots, or by inverting the blending operator. Seismic interferometry and the simultaneous‐source method are related. In this paper we make this relation explicit by deriving deblending as a form of seismic interferometry by MDD. Moreover, we discuss a deblending algorithm for blended data acquired at the surface.

Marine controlled-source electromagnetic interferometry

In marine Controlled-Source Electromagnetics, a boat tows an electric source, whose signal is travelling on various paths to the receiver stations at the ocean bottom. Unfortunately, the signal does not only travel via the subsurface to the receivers, but also directly through the water and via the air-water interface. Signals travelling on the latter two travelpaths do not contain any information about the subsurface. On the contrary, they cover a possible response from a subsurface reservoir. Therefore, one aims to suppress the signal travelling along those paths. Interferometry by multidimensional deconvolution replaces the overburden by a homogeneous halfspace suppressing any interactions with the air-water interface. Furthermore, the direct field is removed and the source is redatumed to a receiver position. Since interferometry by multidimensional deconvolution is a data-driven method, no information about the ocean or the subsurface is needed, except the material parameters at the receiver level. This thesis investigates the benefits and limitations of interferometry by multidimensional deconvolution applied to marine Controlled-Source Electromagnetic data.

Two‐dimensional controlled‐source electromagnetic interferometry by multidimensional deconvolution: spatial sampling aspects

ABSTRACTWe use numerically modelled data sets to investigate the sensitivity of electromagnetic interferometry by multidimensional deconvolution to spatial receiver sampling. Interferometry by multidimensional deconvolution retrieves the reflection response below the receivers after decomposition of the fields into upward and downward decaying fields and deconvolving the upward decaying field by the downward decaying field. Thereby the medium above the receiver level is replaced with a homogeneous half‐space, the sources are redatumed to the receiver level and the direct field is removed. Consequently, in a marine setting the retrieved reflection response is independent of any effect of the water layer and the air above. A drawback of interferometry by multidimensional deconvolution is a possibly unstable matrix inversion, which is necessary to retrieve the reflection response. Additionally, in order to correctly separate the upward and the downward decaying fields, the electromagnetic fields need to be sampled properly. We show that the largest possible receiver spacing depends on two parameters: the vertical distance between the source and the receivers and the length of the source. The receiver spacing should not exceed the larger of these two parameters. Besides these two parameters, the presence of inhomogeneities close to the receivers may also require a dense receiver sampling. We show that by using the synthetic aperture concept, an elongated source can be created from conventionally acquired data in order to overcome these strict sampling criteria. Finally, we show that interferometry may work under real‐world conditions with random noise and receiver orientation and positioning errors.

Green's function retrieval from noise by multidimensional deconvolution

The correlation of noise at two receivers is approximately proportional to the Green's function between these receivers. The approximation is accurate when the medium is lossless and the noise field is equipartitioned. These assumptions are in practice often violated: the medium of interest is often illuminated from one side only, the sources may be irregularly distributed and losses may be significant. For those situations, the correlation function is proportional to a Green's function with a blurred source. The source blurring is quantified by a so-called point-spread function which, like the correlation function, can be derived from the observed data (i.e., without the need to know the actual sources and the medium). The blurred source can be focused by multidimensionally deconvolving the correlation function for the point-spread function. We illustrate the correlation and deconvolution methods with several examples and discuss the advantages and limitations of both methods.

The potential for extracting the electromagnetic earth response from uncorrelated noise

Thermal electromagnetic radiation from an absorbing medium allows for extracting the response of this medium. In the earth, thermal noise is usually very weak and other forms of electromagnetic noise prevail. Noise below radio frequencies is generated in the atmosphere, ionosphere, and magnetosphere, while cosmic noise generates electromagnetic waves above radio frequencies that reach the earth's surface. The noise requirements are discussed for energy and Lagrangian forms of earth response extraction by correlation and multi-dimensional deconvolution. It is shown that if the uncorrelated noise sources are located outside the earth, only the earth response to incident electromagnetic waves can be extracted. Potential applications are found for ground-penetrating radar, and these are illustrated with numerical examples. For coupled seismic waves and electromagnetic fields in fluid-filled porous media, uncorrelated seismic noise is shown to be sufficient to approximately extract the electro-seismic earth response.