Generalized Born inversion of seismic reflection data

Summary . Born inverse methods give accurate and stable results when the source wavelet is impulsive. However, in many practical applications (reflection seismology) an impulsive source cannot be realized and the inversion needs to be generalized to include an arbitrary source function. In this paper, we present a Born solution to the seismic inverse problem which can accommodate an arbitrary source function and give accurate and stable results. It is shown that the form of the generalized inversion algorithm reduces to a Wiener shaping ***filter, which is solved efficiently using a Levinson recursion algorithm. Numerical examples of synthetic and real field data illustrate the validity of our method.     

Inversion of a normally incident reflection seismogram by the Gopinath–Sondhi integral equation

Summary. The one-dimensional acoustic wave equation has been transformed to two coupled first-order equations whose inverse solution is obtained through application of the Gopinath and Sondhi integral equation. A scattering solution of the Schrödinger wave equation for an explosive source leads us to express the kernel of the Gopinath–Sondhi integral equation in terms of a seismic reflection response. A numerical solution of the integral equation obtained by a trapezoidal rule yields a continuous impedance profile whose derivative has step-like discontinuities. The method is illustrated with computer model studies.     

One-way representations of seismic data

A general one-way representation of seismic data can be obtained by substituting a Green's one-way wavefield matrix into a reciprocity theorem of the convolution type for one-way wavefields. From this general one-way representation, several special cases can be derived.
By introducing a Green's one-way wavefield matrix for primaries , a generalized Bremmer series representation is obtained. Terminating this series after the first-order term yields a primary representation of seismic reflection data. According to this representation, primary seismic reflection data are proportional to a reflection operator, 'modified' by primary propagators for downgoing and upgoing waves. For seismic imaging, these propagators need to be inverted. Stable inverse primary propagators can easily be obtained from a one-way reciprocity theorem of the correlation type.
By introducing a Green's one-way wavefield matrix for generalized primaries , an alternative representation is obtained in which multiple scattering is organized quite differently (in comparison with the generalized Bremmer series representation). According to the generalized primary representation, full seismic reflection data are proportional to a reflection operator, 'modified' by generalized primary propagators for downgoing and upgoing waves. Internal multiple scattering is fully included in the generalized primary propagators {either via a series expansion or in a parametrized way). Stable inverse generalized primary propagators can be obtained from the one-way reciprocity theorem of the correlation type. These inverse propagators are the nucleus for seismic imaging techniques that take the angle-dependent dispersion effects due to fine-layering into account.     

Seismic reflection traveltimes in two-dimensional statistically anisotropic random media

Velocity estimation remains one of the main problems when imaging the subsurface with seismic reflection data. Traveltime inversion enables us to obtain large-scale structures of the velocity field and the position of seismic reflectors. However, as the media currently under study are becoming more and more complex, we need to know the finer-scale structures. The problem is that below a certain range of velocity heterogeneities, deterministic methods become difficult to use, so we turn to a probabilistic approach. With this in view, we characterize the velocity field as a random field defined by its first and second statistical moments. Usually, a seismic random medium is defined as a homogeneous velocity background perturbed by a small random field that is assumed to be stationary. Thus, we make a link between such a random velocity medium (together with a simple reflector) and seismic reflection traveltimes. Assuming that the traveltimes are ergodic, we use 2-D seismic reflection geometry to study the decrease in the statistical traveltime fluctuations as a function of the offset (the source–receiver distance). Our formulae are based on the Rytov approximation and the parabolic approximation for acoustic waves. The validity and the limits are established for both of these approximations in statistically anisotropic random media. Finally, theoretical inversion procedures are developed for the horizontal correlation structure of the velocity heterogeneities for the simplest case of a horizontal reflector. Synthetic seismograms are then computed (on particular realizations of random media) by simulating scalar wave propagation via finite difference algorithms. There is good agreement between the theoretical and experimental results.     

The fine structure of the normal incidence synthetic seismogram

Summary . The spectral function of a perfectly elastic, horizontally stratified medium has been demonstrated previously to provide an attractive formulation to describe the properties of the one-dimensional synthetic seismogram (Robinson & Treitel). Here we examine the mathematical framework of the Model in still greater detail. Knowledge of this fine structure of the synthetic seismogram leads to the solution of two particular seismic inverse problems. First, we consider a layered medium with an arbitrary surface reflection coefficient c _o, where | c _o|<1, and which contains an impulsive source immediately above the surface. Given the corresponding synthetic seismogram, we develop an inverse, or backward recursion formalism which recovers the entire series of original reflection coefficients. Second, we consider a similar problem for an impulsive source located just below the surface. Both inversion procedures constitute a continuation of the work of Goupillaud and of Sherwood & Trorey, and represent a generalization of the classical technique originally proposed by Kunetz which, however, only holds for the marine case, c_o =±1. The present approach is not so constrained and thereby becomes applicable to land seismograms as well.
If products of third or higher order in the reflection coefficients can be neglected, significant simplifications arise in the theory. In that event the usual representation of the synthetic seismogram as a ratio of two polynomials in the complex variable z becomes particularly revealing. The numerator polynomial is then approximately equal to the z transform of the reflection coefficient series, while the denominator polynomial is approximately equal to the z transform of the autocorrelation of these reflection coefficients. The resulting simplified theory affords important computational savings in the appropriate backward recursion algorithms.     

Three-dimensional inversion without blocks

Summary. We propose a method for solving non-linear inverse problems in the case where the unknown is a function of the spatial coordinates and the data set is discrete and finite. The method is based on a generalized leastsquares criterion, it is defined directly for non-linear problems (without previous linearization of the forward problem), and in the particular linear case it gives the same results (although slightly more general) than the Backus & Gilbert approach. As an example, we apply the method to the three-dimensional seismic velocity inverse problem, using as data the arrival times of seismic waves. The following paper (Nercessian et al .) shows some esults obtained using the present method.     

Weak-contrast reflection/transmission coefficients in weakly anisotropic elastic media: P-wave incidence

We present approximate displacement and energy PP and PS reflection/transmission coefficients for weak-contrast interfaces in general weakly anisotropic elastic media. The coefficients were obtained by applying first-order perturbation theory and then expressed in a compact and relatively simple form. The formulae can be used for arbitrary orientations of the incidence plane and interface, without the need to transform the elasticity parameters to a local Cartesian coordinate system. The accuracy of the approximate formulae is illustrated for the PS reflection coefficient for two synthetic models. For these models, we also study the possibility of using the approximate PP reflection coefficient in the inverse problem.     

Full reconstruction of a layered elastic medium from P-SV slant-stack data

Summary. We develop a méthod of reconstructing the elastic paraméters as functions of depth, for a horizontally stratified, isotropic elastic half-space. Unlike previous schemes, which have been able to retrieve the shear wave speed and density from SH seismograms slant stacked at two angles, our méthod makes use of P - SV data at a single stacking paraméter to obtain all three elastic constants. The data required are the elements of the full reflection matrix at the surface, corresponding to measurements of two separate components of the response to two independent sources, one explosive, the other generating shear waves.
In developing this inverse scheme fundamental differences emerge between the acoustic or SH problem, and the coupled P - SV case, the most important being in the nature of the interfacial scattering matrix. We show that it is not possible to make use of the downward reflection data for an interface to determine directly the remaining reflection and transmission coefficients, but that the scattering data may be completed by applying a simple iterative procedure at each interface.
We show the result of applying our inverse scheme to seismograms generated for a six-layered model, including a low-velocity layer. We are able to reconstruct both wave speeds and the density as functions of depth, all quantities being in close agreement with the original model.     

Statistical analysis of spatial point patterns on deep seismic reflection data: a preliminary test

Spatial point patterns generated from bitmaps of images of processed reflection seismic profiles are analysed to quantify the reflectivity patterns. The point process characteristics for two different regions of a deep seismic reflection profile in northwestern Canada demonstrate that in both cases the points are not randomly distributed and that the point pattern distribution is different between the regions. The cluster effects for small point pair distances are stronger for the region of data where there is strong sedimentary layering than for the region where the layering is less distinct. As a result, it appears that future developments in point pattern analysis may provide a new tool for analysing spatial variations in reflection data.     

Deep crustal geoelectric structure beneath the Northumberland Basin

Summary. In terms of lateral variations in conductivity structure, the southern Southern Uplands and Northumberland Basin are characterized by a region of attenuated vertical magnetic fields with small spatial gradients reflecting the presence of a substantial conducting zone. Five magnetotelluric data sets from the region have been analysed to provide accurate and unbiased estimates of the impedance tensor. The response data are used to investigate the deep geoelectric crustal structure of the region. Three appropriate sets of response data have been subjected to two construction algorithms for 1-D inversion. The geoelectric profiles recovered identify a deep crustal conducting zone underlying the Northumberland Basin. The zone, modelled as a layered structure, dips steeply from mid-crustal depths underneath the Northumberland Basin to lower crustal depths to the NW. The structure thus correlates, in location and geometry, with a deep crustal reflecting wedge detected offshore by a deep seismic reflection profile.     

Joint traveltime inversion of wide-angle seismic data and a deep reflection profile from the central North Sea

We report results from the Seismic Wide-Angle and Broadband Survey carried out over the Mid North Sea High. This paper focuses on integrating the information from a conventional deep multichannel reflection profile and a coincident wide-angle profile obtained by recording the same shots on a set of ocean bottom hydrophones (OBH). To achieve this integration, a new traveltime inversion scheme was developed (reported elsewhere) that was used to invert traveltime information from both the wide-angle OBH records and the reflection profile simultaneously. Results from the inversion were evaluated by producing synthetic seismograms from the final inversion model and comparing them with the observed wide-angle data, and an excellent match was obtained. It was possible to fine-tune velocities in less well-resolved parts of the model by considering the critical distance for the Moho reflection. The seismic velocity model was checked for compatibility with the gravity field, and used to migrate and depth-convert the reflection profile. The unreflective upper crust is characterized by a high velocity gradient, whilst the highly reflective lower crust is associated with a low velocity gradient. At the base of the crust there are several subhorizontal reflectors, a few kilometres apart in depth, and correlatable laterally for several tens of kilometres. These reflectors are interpreted as representing a strike section through northward-dipping reflectors at the base of the crust, identified on orthogonal profiles by Freeman et al. (1988) as being slivers of subducted and imbricated oceanic crust, relics of the mid-Palaeozoic Iapetus Ocean.     

Modelling and inversion of single-ended refraction data from the shot gathers of multifold deep seismic reflection profiling—an approach for deriving the shallow velocity structure

A multifold crustal-scale deep seismic near-vertical reflection profile generates a large number of single-ended shot gathers, which provide redundant data sets because of overlapping coverage of the shallow refractors. We present an approach for deriving the shallow velocity structure by modelling and inversion of single-ended seismic refraction first arrival traveltime data. We apply this method to a data set acquired with a 12-km long spread with 100 m spacing of shots and receivers, of the Neoproterozoic Marwar basin in the NW Indian shield. The approach is shown to be quite successful for delineating the shallow refractor depths, steep dips and velocities, even in the absence of regular reverse refraction profiles. The study reveals two-layered sedimentary formations, Malani volcanics and a complicated basement configuration of the Marwar basin, and provides a measure of resolution and uncertainty of the estimated model parameters. A seismic section of the near-trace gather is found to be qualitatively consistent with the derived structural features of the basin. The relative highs and lows, observed in the Bouguer gravity profile, further corroborate the derived velocity model. The present approach can be especially useful in offshore areas and elsewhere, where the single-ended multifold seismic profiles are the only available data sets.     

Southwest Barents Sea rift basin evolution: comparing results from backstripping and time‐forward modelling

We present tectonic models of progressive basin formation in the south‐west Barents Sea derived as part of the PETROBAR project (Petroleum‐related studies of the Barents Sea region). The basin architecture developed as a multi‐stage rift preceding the creation of the sheared/transtensional margin conjugate to NE Greenland. N‐ to NNE‐striking basins, with sediment thicknesses in places exceeding 15 km, are separated by basement highs. We use two basin analysis approaches, BMT? backstripping and TecMod?time‐forward modelling, to determine stretching factors through time along the profile PETROBAR‐07. This 550 km‐long profile derived from wide‐angle reflection/refraction seismic data acquired in 2007, coincident with deep multichannel seismic reflection data. Detailed stratigraphic analysis of the reflection profile, in concert with a dense grid of 2D profiles tied to wells, provides timing and water depth constraint for the models. Velocity analysis of the wide‐angle data provides constraint on the cumulative crustal stretching. The north‐west trending cross‐section extends from continental craton, at the Varanger Peninsula, to within 16 km of the interpreted continent–ocean boundary. Rifting along the profile was episodic, with four distinct phases of basin formation during the Carboniferous, the Late Permian–Triassic, the Late Jurassic–Early Cretaceous and the Late Cretaceous–Eocene. Collectively, the basins exhibit a general trend of younging, narrowing, and deepening oceanward, suggesting a gradual focusing of rifting prior to final breakup. Cumulative stretching factors derived from BMT and TecMod correlate well with observed crustal thinning, and the two models provide uncertainty bounds for stretching factors for the separate rift phases. In contrast to orthogonally rifted margins, stretching is relatively minor immediately prior to transform breakup, with greater stretching occurring during earlier rift phases.     

A Bayesian approach to inverse modelling of stratigraphy, part 1: method

The inference of ancient environmental conditions from their preserved response in the sedimentary record still remains an outstanding issue in stratigraphy. Since the 1970s, conceptual stratigraphic models (e.g. sequence stratigraphy) based on the underlying assumption that accommodation space is the critical control on stratigraphic architecture have been widely used. Although these methods considered more recently other possible parameters such as sediment supply and transport efficiency, they still lack in taking into account the full range of possible parameters, processes, and their complex interactions that control stratigraphic architecture. In this contribution, we present a new quantitative method for the inference of key environmental parameters (specifically sediment supply and relative sea level) that control stratigraphy. The approach combines a fully non‐linear inversion scheme with a ‘process–response’ forward model of stratigraphy. We formulate the inverse problem using a Bayesian framework in order to sample the full range of possible solutions and explicitly build in prior geological knowledge. Our methodology combines Reversible Jump Markov chain Monte Carlo and Simulated Tempering algorithms which are able to deal with variable‐dimensional inverse problems and multi‐modal posterior probability distributions, respectively. The inverse scheme has been linked to a forward stratigraphic model, BARSIM (developed by Joep Storms, University of Delft), which simulates shallow‐marine wave/storm‐dominated systems over geological timescales. This link requires the construction of a likelihood function to quantify the agreement between simulated and observed data of different types (e.g. sediment age and thickness, grain size distributions). The technique has been tested and validated with synthetic data, in which all the parameters are specified to produce a ‘perfect’ simulation, although we add noise to these synthetic data for subsequent testing of the inverse modelling approach. These tests addressed convergence and computational‐overhead issues, and highlight the robustness of the inverse scheme, which is able to assess the full range of uncertainties on the inferred environmental parameters and facies distributions.     

Interpretation of subhorizontal crustal reflections by metamorphic and rheologic effects in the eastern part of the Pannonian Basin

The geologic origin of subhorizontal reflections, often observed in crustal seismic sections, was investigated by establishing metamorphic facies and strength of rocks in depth, and correlating these properties to seismic reflection sections from eastern Hungary. Estimation of the depths of metamorphic mineral stability zones utilized the principles developed by Fyfe et al. and known geothermal data of the area. The strength versus depth profile was derived by relating local seismic P -wave interval velocities to Meissner et al. 's activation energy. The results show that the series of subhorizontal reflections, observed in the Pannonian Basin, are a consequence of combined metamorphic and rheologic changes in depths. The synthesis of the integrated data set suggests that the retrograde alteration of the pre-Tertiary basement above the percolation threshold was made possible by the softening effect of shear zones and their water-conducting capacity. The subhorizontal reflections of highest energy, of the consolidated crust below the percolation threshold, originate in the depths of greenschist, amphibolite and granulite metamorphic mineral facies, which were formed in geothermal and pressure conditions similar to those existing today. These results imply the overprint of earlier (Variscan) metamorphic sequences of the crust by more recent retrograde metamorphic processes.     

Local interpolation using a distributed parallel supercomputer

Abstract

Large spatial interpolation problems present significant computational challenges even for the fastest workstations. In this paper we demonstrate how parallel processing can be used to reduce computation times to levels that are suitable for interactive interpolation analyses of large spatial databases. Though the approach developed in this paper can be used with a wide variety of interpolation algorithms, we specifically contrast the results obtained from a global ‘brute force’ inverse–distance weighted interpolation algorithm with those obtained using a much more efficient local approach. The parallel versions of both implementations are superior to their sequential counterparts. However, the local version of the parallel algorithm provides the best overall performance.     

Full three-dimensional tomography: a comparison between the scattering-integral and adjoint-wavefield methods

This paper analyses the computational issues of full 3-D tomography, in which the starting model as well as the model perturbation is 3-D and the sensitivity (Fréchet) kernels are calculated using the full physics of 3-D wave propagation. We compare two formulations of the structural inverse problem: the adjoint-wavefield (AW) method, which back-propagates the data from the receivers to image structure, and the scattering-integral (SI) method, which sets up the inverse problem by calculating and storing the Fréchet kernels for each data functional. The two inverse methods are closely related, but which one is more efficient depends on the overall problem geometry, particularly on the ratio of sources to receivers, as well as trade-offs in computational resources, such as the relative costs of compute cycles to data storage. We find that the SI method is computationally more efficient than the AW method in regional waveform tomography using large sets of natural sources, although it requires more storage.     

盐湖化学分析测试中心

浅地层地震剖面探测技术是一种基于声学原理连续走航式探测水下浅部地层结构与构造、沉积物特征及矿物分布的地球物理探测方法,具有连续性好、分辨率高和直观性强的特点。在湖泊沉积尤其是水下沉积研究中,弥补了沉积物钻孔分析"一柱之见"的局限性,能够更加直观准确地揭示湖泊沉积层垂向和横向地层结构、构造及其空间分布。回顾浅地层地震剖面探测技术的发展历程,基于湖泊地震相解释的原理,其在湖泊水位、湖泊古地质活动、湖盆形态演变等研究中已有一定的应用,初显了浅地层地震剖面探测技术在晚更新世以来湖泊沉积研究中的独特优势。近期,与RS和GIS技术结合,实现了湖泊沉积2D和3D可视化。湖泊水浅、植物多、浮泥厚等因素影响反射波与图像质量,浅地层地震剖面探测技术在硬件设备、数据采集时的参数设置及数据后处理软件功能完善等方面,仍有待提高。     

Sequential integrated inversion of refraction and wide-angle reflection traveltimes and gravity data for two-dimensional velocity structures

A new algorithm is presented for the integrated 2-D inversion of seismic traveltime and gravity data. The algorithm adopts the 'maximum likelihood' regularization scheme. We construct a 'probability density function' which includes three kinds of information: information derived from gravity measurements; information derived from the seismic traveltime inversion procedure applied to the model; and information on the physical correlation among the density and the velocity parameters. We assume a linear relation between density and velocity, which can be node-dependent; that is, we can choose different relationships for different parts of the velocity–density grid. In addition, our procedure allows us to consider a covariance matrix related to the error propagation in linking density to velocity. We use seismic data to estimate starting velocity values and the position of boundary nodes. Subsequently, the sequential integrated inversion (SII) optimizes the layer velocities and densities for our models. The procedure is applicable, as an additional step, to any type of seismic tomographic inversion.
We illustrate the method by comparing the velocity models recovered from a standard seismic traveltime inversion with those retrieved using our algorithm. The inversion of synthetic data calculated for a 2-D isotropic, laterally inhomogeneous model shows the stability and accuracy of this procedure, demonstrates the improvements to the recovery of true velocity anomalies, and proves that this technique can efficiently overcome some of the limitations of both gravity and seismic traveltime inversions, when they are used independently.
An interpretation of field data from the 1994 Vesuvius test experiment is also presented. At depths down to 4.5 km, the model retrieved after a SII shows a more detailed structure than the model obtained from an interpretation of seismic traveltime only, and yields additional information for a further study of the area.