Arie Verdel

Tutorial on seismic interferometry: Part 1 — Basic principles and applications

Seismic interferometry involves the crosscorrelation of responses at different receivers to obtain the Green’s function between these receivers. For the simple situation of an impulsive plane wave propagating along the x-axis, the crosscorrelation of the responses at two receivers along the x-axis gives the Green’s function of the direct wave between these receivers. When the source function of the plane wave is a transientas in exploration seismology or a noise signalas in passive seismology, then the crosscorrelation gives the Green’s function, convolved with the autocorrelation of the source function. Direct-wave interferometry also holds for 2D and 3D situations, assuming the receivers are surrounded by a uniform distribution of sources. In this case, the main contributions to the retrieved direct wave between the receivers come from sources in Fresnel zones around stationary points. The main application of direct-wave interferometry is the retrieval of seismic surface-wave responses from ambient noise and the subsequent tomographic determination of the surfacewave velocity distribution of the subsurface. Seismic interferometry is not restricted to retrieving direct waves between receivers. In a classic paper, Claerbout shows that the autocorrelation of the transmission response of a layered medium gives the plane-wave reflection response of that medium. This is essentially 1D reflected-wave interferometry. Similarly, the crosscorrelation of the transmission responses, observed at two receivers, of an arbitrary inhomogeneous medium gives the 3D reflection response of that medium. One of the main applications of reflected-wave interferometry is retrieving the seismic reflection response from ambient noise and imaging of the reflectors in the subsurface. A common aspect of direct- and reflected-wave interferometry is that virtual sources are created at positions where there are only receivers without requiring knowledge of the subsurface medium parameters or of the positions of the actual sources.

Reflection images from ambient seismic noise

One application of seismic interferometry is to retrieve the impulse response (Greens function) from crosscorrelation of ambient seismic noise. Various researchers show results for retrieving the surface-wave part of the Greens function. However, reflection retrieval has proven more challenging. We crosscorrelate ambient seismic noise, recorded along eight parallel lines in the Sirte basin east of Ajdabeya, Libya, to obtain shot gathers that contain reflections. We take advantage of geophone groups to suppress part of the undesired surface-wave noise and apply frequency-wavenumber filtering before crosscorrelation to suppress surface waves further. After comparing the retrieved results with data from an active seismic exploration survey along the same lines, we use the retrieved reflection data to obtain a migrated reflection image of the subsurface.

Retrieval of reflections from seismic background‐noise measurements

The retrieval of the earths reflection response from cross?correlations of seismic noise recordings can provide valuable information, which may otherwise not be available due to limited spatial distribution of seismic sources. We cross?correlated ten hours of seismic background?noise data acquired in a desert area. The cross?correlation results show several coherent events, which align very well with reflections from an active survey at the same location. Therefore, we interpret these coherent events as reflections. Retrieving seismic reflections from background?noise measurements has a wide range of applications in regional seismology, frontier exploration and long?term monitoring of processes in the earths subsurface.

Estimation of the effect of nonisotropically distributed energy on the apparent arrival time in correlations

Correlations of random seismic noise are now widely used to retrieve the Green’s function between two points. Whereas this technique provides useful results in tomography and monitoring studies, it is mainly limited by an uneven distribution of noise sources. In that case, theoretical requirements are not completely fulfilled and we may wonder how reliable the reconstructed signals are, in particular for the purpose of estimating traveltime from correlations. This study finds a way to quantify effects of a nonisotropic noise field by estimating the arrival-time error resulting from a particular nonisotropic distribution of recorded wave intensity. Our study is based on a theoretical prediction of this bias and we successfully test the theory by comparing the theoretical expectation to real measurements from seismic-prospecting data. In particular, we distinguish between the effects of source distribution and the effects of medium heterogeneity between the sources and the region of receivers. We find relat...

A microlocal analysis of migration

Abstract The mathematics of the propagation of seismic energy relevant in seismic reflection experiments is assumed to be governed by the linear acoustic wave equation, in which the coefficient is the speed of sound in the subsurface. If the speed of sound suddenly changes, the seismic signal is (partly) reflected. To find the position of these changes one considers first the so-called forward map, which sends the coefficient of the wave equation to its solution at the surface of the earth, where the recording equipment is positioned. This map is highly non-linear. For inversion one therefore usually takes its formal derivative, which leads to a linearized inverse problem. It is well known that this linearized forward map corresponds in a high frequency approximation to a Fourier integral operator. The construction of a parametrix for this Fourier integral operator requires the computation of the so-called normal operator, i.e. the composition of the linearized forward map with its adjoint. An important result by G. Beylkin [The inversion problem and applications of the generalized Random transform, Comm. Pure Appl. Math. 37 (1984) 579–599] states that, if there are no caustics in the medium, this normal operator is an elliptic pseudo-differential operator. In many practical situations, however, the no-caustics assumption is violated. In this paper a microlocal analysis of this more general case will be presented. We will show that for spatial dimensions less than or equal to 3, the normal operator remains a Fourier integral operator, be it not a pseudo-differential operator anymore. Instead, it is the sum of an elliptic pseudo-differential operator and a more general Fourier integral operator of lower order than the pseudo-differential part. We will also formulate a mild injectivity condition on the traveltime function under which Beylkins result remains true, i.e. under which the normal operator is purely pseudo-differential. This injectivity condition includes various types of multi-valued traveltimes, which occur frequently in practice. Its geometrical interpretation will be discussed. Finally, we derive an approximate explicit formula for the inverse, which is suitable for numerical evaluation.

Depth migration by the Gaussian beam summation method

Seismic depth migration aims to produce an image of seismic reflection interfaces. Ray methods are suitable for subsurface target-oriented imaging and are less costly compared to two-way wave-equation-based migration, but break down in cases when a complex velocity structure gives rise to the appearance of caustics. Ray methods also have difficulties in correctly handling the different branches of the wavefront that result from wave propagation through a caustic. On the other hand, migration methods based on the two-way wave equation, referred to as reverse-time migration, are known to be capable of dealing with these problems. However, they are very expensive, especially in the 3D case. It can be prohibitive if many iterations are needed, such as for velocity-model building. Our method relies on the calculation of the Green functions for the classical wave equation by per-forming a summation of Gaussian beams for the direct and back-propagated wavefields. The subsurface image is obtained by cal-culating ...

Convergence of the two-point correlation function toward the Green's function in the context of a seismic-prospecting data set

When considering direct waves in the correlation process, the Green’s function is reconstructed when using an even distribution of seismic sources or when the source distribution is restricted to the direction close to the alignment of the sensors. On the other hand, when considering records of coda waves, the convergence is achieved for any source distribution, as expected theoretically. We extract the expected amplitude decay along a seismic profile from the correlation functions when an even distribution of sources is considered or when the time window includes scattered waves.

Detection of lateral velocity contrasts by crosswell traveltime tomography

Crosswell traveltime tomography is a common technique in the oil industry for determining the velocity function in the plane between two boreholes. However, the method suffers from the well‐known problem that the lateral resolution is far less than the vertical resolution because of the unfavorable illumination conditions for survey geometries comprising vertical wells. Consequently, it is very difficult to image sudden lateral changes in the velocity function accurately using this technique. We propose a method for determining such changes, which severely constrains the solution space by inverting for the position of a lateral velocity contrast only. The velocity model on each side of the contrast is derived from the well logs. The potential of the method is first demonstrated in two synthetic examples in which its properties are compared with those of an unconstrained 2-D tomographic inversion. As expected, the constrained method has much better convergence properties than the unconstrained one in these...

Source depopulation potential and surface-wave tomography using a crosscorrelation method in a scattering medium

We use seismic prospecting data on a 40 � 40 regular grid of sources and receivers deployed on a 1 km � 1 km area to assess the feasibility and advantages of velocity analysis of the shallow subsurface by means of surface-wave tomography with Green’s functions estimated from crosscorrelation. In a first application we measure Rayleigh-wave dispersion curves in a 1D equivalent medium. The assumption that the medium is laterally homogeneous allows using a simple projection scheme and averaging of crosscorrelation functions over the whole network. Because averaging suppresses noise, this method yields better signal-to-noise ratio than traditional active-source approaches, and the improvement can be estimated a priori from acquisition parameters. We find that high-quality dispersion curves can be obtained even when we reduce the number of active sources used as input for the correlations. Such source depopulation can achieve significant reduction in the cost of active source acquisition. In a second application we compare Rayleigh-wave group velocity tomography from raw and reconstructed data. We can demonstrate that the crosscorrelation approach yields group velocity maps that are similar to active source maps. Scattering has an importance here as it may enhance the crosscorrelation performance. We quantify the scattering properties of the medium using mean free path measurements from coherent and incoherent parts of the signal. We conclude that for first-order velocity analysis of the shallow subsurface, the use of crosscorrelation offers a cost-effective alternative to methods that rely exclusively on active sources.

Event-driven Seismic Interferometry with Ambient Seismic Noise

By cross-correlating recordings of ambient seismic noise, one can retrieve the subsurface reflection response. The quality of the retrieved reflections would depend on the qualities of the ambient noise. In a previous study, we cross-correlated ambient-noise data recorded in a desert area in North Africa and showed that we retrieved reflections. This was done assuming that body-wave noise continuously illuminates the recording array. But this is not necessarily true - noise which carries body-wave information can be present only at certain times. We now use only parts of the recorded noise during the correlation process. These parts contain identifiable body-wave events. We show that the results, retrieved only from the noise containing the events, exhibit clearer reflection arrivals.