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S U M M A R YA migration algorithm based on the least-squares formulation will find the correct reflector amplitudes if proper migration weights are applied. The migration weights can be viewed as a pre-conditioner for a gradient-based optimization problem. The pre-conditioner should approximate the pseudo-inverse of the Hessian of the least-squares functional. Usually, an infinite receiver coverage is assumed to derive this approximation, but this may lead to poor amplitude estimates for deep reflectors.To avoid the assumption of infinite coverage, new amplitude-preserving migration weights are proposed based on a Born approximation of the Hessian. The expressions are tested in the context of frequency-domain finite-difference two-way migration and show improved amplitudes for the deeper reflectors.Migration algorithms produce the depth locations and relative amplitude behaviour of reflectors in the Earth from measured seismic data. The classic imaging principle (Claerbout 1971; Esmersoy 1986) for shot-based migration states that reflectors are located where the forwardpropagated wavefield from the source correlates with the backward-propagated wavefield of the receiver data. To find a suitable approximation for the reflector amplitudes, the correlation is divided by the square of the incident field. This approach leads, among others, to the classic reversed-time migration algorithm (Baysal et al. 1983).In the 1980s it was observed that the imaging principle can be obtained by formulating migration as an inverse problem based on a least-squares functional (Lailly 1983; Tarantola 1984;Beylkin 1985). This opened up research on true-amplitude or amplitude-preserving migration. Beylkin (1985) derived an amplitude-preserving migration algorithm by viewing migration as an inverse Radon transform under the assumption of infinite and continuous receiver coverage using high-frequency asymptotes. Lailly (1983) and Tarantola (1984) started with a least-squares error functional that measures the difference between observed and synthetic data. They found that the gradient of this functional with respect to the underlying model corresponds to a migration image. This approach has the advantage that it can be used independently of the numerical scheme that models the synthetic data and independently of the acquisition geometry. In this setting, amplitude-preserving migration is obtained by scaling the gradient by the pseudo-inverse of the Hessian of the least-squares functional. The least-squares approach differs from the imaging principle approach or Beylkin's approach because it can be based on the full data set, resulting in migration weights that are different for multishot multi-offset data. Also, irregular acquisition geometries can be taken into account.The main difficulty with the least-squares approach is the size of the Hessian, the matrix of the second derivatives of the error functional with respect to the model parameters, which is too large to be directly used in practical applications because an approximation of...

[1] The retrieval of the earth's 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 earth's subsurface.

S U M M A R YWave-equation traveltime tomography tries to obtain a subsurface velocity model from seismic data, either passive or active, that explains their traveltimes. A key step is the extraction of traveltime differences, or relative phase shifts, between observed and modelled finite-frequency waveforms. A standard approach involves a correlation of the observed and measured waveforms. When the amplitude spectra of the waveforms are identical, the maximum of the correlation is indicative of the relative phase shift. When the amplitude spectra are not identical, however, this argument is no longer valid. We propose an alternative criterion to measure the relative phase shift. This misfit criterion is a weighted norm of the correlation and is less sensitive to differences in the amplitude spectra. For practical application it is important to use a sensitivity kernel that is consistent with the way the misfit is measured. We derive this sensitivity kernel and show how it differs from the standard banana-doughnut sensitivity kernel. We illustrate the approach on a cross-well data set.In ray-based tomography, the aim is to construct a subsurface velocity model that explains the picked traveltimes of the measured data. Such a model can be obtained in an iterative manner by back projecting the traveltime differences along rays in the current velocity model. This procedure will lead to satisfactory results when the wave propagation is sufficiently well approximated by ray theory. To extract more information from the data than just the traveltimes of a few selected arrivals, seismologists are moving towards full-waveform processing and inversion of all available data. This trend is driven by the availability of high-quality broad-band data (earthquake data from USArray, for example) and a need to incorporate finite-frequency effects to process data from geologically complex areas (sub-salt exploration for the detection of hydrocarbons, for example). Also, the computing resources needed to routinely model 3-D wave-propagation in complex media are becoming a commodity. There have been two major developments regarding finite-frequency or full-waveform inversion in the last decades. The first is least-squares inversion-or waveform tomography-of seismic data pioneered by Tarantola & Valette (1982) that was aimed originally at inversion of active seismic reflection data. The second is the development of a finite-frequency analogue of ray-based traveltime tomography by Luo & Schuster (1991) (see also TrompRe-use of this article is permitted in accordance with the Terms and Conditions set out at

The goal of this study is a better understanding of the numerous sound propagation mechanisms in granular materials. In a static, regular (crystal), 3D packing, a small perturbation is created on one side and examined during its propagation through frictionless and frictional packings. The perturbation can be applied in longitudinal and shear direction in order to excite different modes of information propagation, including rotational modes as well. Wave speed and dispersion relation derived from simulation data are compared to those given by a theoretical approach based on a micro-macro transition. The detailed analysis of the wave velocity reveals an interesting acceleration close to the source. Finally a step towards real packings is made by introducing either friction or a tiny (but decisive) polydispersity in the particle size.

Results for wave-equation migration in the frequency domain using the constant-density acoustic two-way wave equation have been compared to images obtained by its one-way approximation. The two-way approach produces more accurate reflector amplitudes and provides superior imaging of steep flanks. However, migration with the two-way wave equation is sensitive to diving waves, leading to low-frequency artifacts in the images. These can be removed by surgical muting of the input data or iterative migration or high-pass spatial filtering. The last is the most effective.Iterative migration based on a least-squares approximation of the seismic data can improve the amplitudes and resolution of the imaged reflectors. Two approaches are considered, one based on the linearized constantdensity acoustic wave equation and one on the full acoustic wave equation with variable density. The first converges quickly. However, with our choice of migration weights and high-pass spatial filtering for the linearized case, a real-data migration result shows little improvement after the first iteration. The second, nonlinear iterative migration method is considerably more difficult to apply. A real-data example shows only marginal improvement over the linearized case.In two dimensions, the computational cost of the twoway approach has the same order of magnitude as that for the one-way method. With our implementation, the two-way method requires about twice the computer time needed for one-way wave-equation migration.

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