Seismic Traveltime Tomography Research Articles

A neural network for weighted least-squares criteria of traveltime tomography

TH network has been applied successfully to linear optimum and some quadratic optimum problems. This paper discusses how to determine the gain function of the net amplifiers and choose appropriate parameters to solve weighted least-squares problems. To test the performance of the net, it is used for seismic traveltime tomography in computer simulation. Two media of different contrast are taken in simulation. The experimental results show that if the parameters are determined appropriately, the performance of the network is good and the results are close to the ideal ones.

Three‐dimensional seismic image of a geothermal reservoir: The Geysers, California

Three‐dimensional seismic travel‐time tomography of The Geysers geothermal area, in the coast ranges of northern California, shows a strong (−9%) anomaly in VP/VS, the ratio of the compressional and shear wave speeds, that is not evident in VP alone and corresponds closely to the most intensively exploited part of the geothermal reservoir. This anomaly probably indicates low pore pressure and relatively dry conditions, caused partly by boiling of pore water as steam is extracted. Steam pressure decreases over the last decade have probably caused seismologically measurable changes in wave speeds. Tomographic measurement of VP/VS is a promising technique both for identifying geothermal resources and for monitoring them during exploitation.

Relative-error-based non-linear inversion: application to seismic traveltime tomography

SUMMARY The aim of geophysical inversion is to infer information about the subsurface model parameters subject to certain constraints and plausibility considerations. It is achieved through a systematic adjustment of model parameters along a solution path to the minimum of an objective function. The objective function is related to both the observations (data) and the a priori information about the subsurface model parameters (model). When different statistics are assumed for the data and the a priori information about the model, different objective functions are constructed and different solution models become inevitable. Traditionally, data residuals are used directly in constructing the objective function. If the residuals are statistically proportional to the data magnitudes, a least-squares (LS) method often leads to a solution model biased towards (i.e. which emphasizes) observations of large magnitude, whereas a least absolute deviation (LAD) method can reduce this datamagnitude-dependent effect. An alternative is to formulate the objective function in terms of relative differences. It completely eliminates the possible solution bias due to the intrinsic magnitude-dependent discrepancies in the objective function. The solution path is altered in such a way that.data residuals are reduced in a more uniform manner. The idea is illustrated through a seismic traveltime tomographic inversion. Traveltime residuals are often correlated with ray-path lengths. Greater residuals of long ray paths dominate the objective function in the traditional tomographic formulation. The solution path is thus strongly affected by those long ray paths, and short ray paths play a reduced role in the inversion. The final solution is biased toward longer ray paths. The new alternative formulation eliminates this bias by exploiting the relative-error measure to give equal weights to data of the same quality regardless of their ray-path lengths.

ジオ・トモグラフィの基礎

The “Geo-Tomography” is now an essential tool to study geo-sciences.There are severalapplications of tomography in geophysics. e. g., seismology, geodesy, geomagnetism, gravity, etc. The tomography using cosmic ray, “Muons”, is a new application. In this paper, theauthor intends to give a general view of the geo-tomography and present the relation amongdifferent methods. Detailed description of each application is described by different authors inthis special volume.There are several different methods in tomography: Matrix method, ART or SIRT, CG, backprojection, backpropagation, p-tau trans form and Fourier method. These are overlappedeach other by some extents. The tomography initiated in the medical use and it is called “C omputed Tomography”. In the medical use, sets of (X-ray) parallel projections along lineswith a constant angle to the reference axis are used. In the filtered backprojection method, the convolution of the result of parallel projection to filter-function is integrated over wholeprojection angles. In the contrast with the medical projection method, geophysical applicationsof tomography tend to be limited in narrow projection angles. This narrow aperture will blurthe real image.Several methods have been used in the geo-tomography. DLSQ, SIRT and CG are themost powerful tools for traveltime tomography such as crosshole-VSP tomography and seismictravel time tomography. Recently, waveform inversion using high-frequency seismic wavessuggested its powerfulness to obtain the fine structure of the earth. However, since the waveforminversion has a nature of strong nonlinearity, it is necessary to use traveltime tomography withor in advance of the waveform inversion.

既知の情報を用いた制約付き速度再構成による弾性波トモグラフィ解析の精度の向上

既知の情報を用いた制約付き速度再構成による弾性波トモグラフィ解析の精度の向上

Linearization of the eikonal equation

Seismic traveltime tomography is a nonlinear inverse problem wherein an unknown slowness model is inferred from the observed arrival times of seismic waves. Nonlinearity arises because the raypath connecting a given source and receiver depends on the slowness. Specifically, if L(s) designates a raypath through the slowness model s between two fixed endpoints, then the path integral for traveltime [Formula: see text] is a nonlinear functional of s because it does not, in general, satisfy the superposition condition (i.e., [Formula: see text] where [Formula: see text] and [Formula: see text] are two different slowness models). The tomographic inverse problem can be solved after linearizing the traveltime expression about a known slowness model [Formula: see text]. This linearized expression is usually obtained by appealing to Fermat’s principle (e.g., Nolet, 1987). Alternately, the required relation can be rigorously derived via ray‐perturbation theory (Snieder and Sambridge, 1992). The purpose of this note is to present a straightforward derivation of the same result by linearizing the eikonal equation for traveltimes. Wenzel (1988) adopts this approach, but his method of proof cannot be generalized to heterogeneous 3-D media. A full 3-D treatment is given here. The proof is remarkably simple, and thus it is quite possible that others have discovered it previously.

Non-linear optimization for seismic traveltime tomography

This paper presents a non-linear algorithmic approach for seismic traveltime. It is based on large-scale optimization using non-linear least-squares and trust-region methods. These methods provide a natural way to stabilize algorithms based on Newton's iteration for non-linear minimization. They also correspond to an alternative (and often more efficient) view of the Levenberg-Marquardt method. Numerical experience on synthetic data and on real borehole-to-borehole problems are presented. In particular, results produced by the new algorithm are compared with those of Ivansson (1985) for the Kra˙kema˙la experiment

Velocity model inversion using parametric curves

SUMMARY The use of parametric curves to define velocity models in seismic traveltime tomography is proposed. This approach provides increased flexibility in shaping the velocity models because it allows inversion for both velocity and grid point adjustments. The method is applied to a data set generated from a 1-D profile having sharp discontinuities. The results are compared with those obtained from other methods currently in use for traveltime tomography. It is found that large gradients and discontinuities can be retrieved with improved accuracy and that instabilities caused by non-optimal prior selection of the inversion grid can be avoided. The results appear to be robust when the sizes of both data and model space are reduced which makes the method appealing for the solution of large-scale tomographic problems.

What patterns of heterogeneity in the Earth's mantle can be revealed by seismic travel time tomography?

We investigate which patterns of seismic heterogeneity in the mantle would be returned reliably by a tomographic inversion in which the model mantle is parametrized by a set of discrete, non-overlapping voxels. We construct synthetic data sets based on real ray sampling of the mantle by introducing spherical harmonic patterns of velocity heterogeneity and perform inversions of the synthetic data. We expand the resulting voxel model in spherical harmonics and compare the power at each degree and in each model layer with the input spherical harmonics in order to determine which patterns produced by inversions of real data may be deemed reliable and to identify patterns which must be viewed with skepticism. We find that while the power input to a particular pattern of heterogeneity in the 0–200 km layer is generally recovered accurately, the pattern itself is poorly determined in this layer. A pattern in the 200–400 km layer is more precisely determined though the power contained in the pattern is consistently underestimated and more leakage occurs to the layers above and below. The transition zone, 400–670 km, shows similarly strong control of lateral heterogeneity patterns but tests return a more accurate estimate of input powre than for the second layer. The l = 2, 4, and 6 components all show accurate recovery in the 400–670 km layer. This result supports previous findings, from inversions with real data, that l = 2 is a significant pattern of heterogeneity in the mantle's transition zone and that l = 4 is not a significant pattern. For the entire upper mantle, l = 6 would be retrieved reliably and its constructive behavior in upper mantle models derived with real data is confirmed. These tests also demonstrate the inability of our inversion procedure to retrieve shorter wavelength features in the lower mantle. The results for our lowermost layer, D″, must be considered suspect due to the inadequate constraints placed on model values by our ray coverage and the sensitivity of these results to noise in the data.

Nonlinear high‐resolution three‐dimensional seismic travel time tomography

A tomographic inversion procedure is described and applied to a synthetic three‐dimensional (3‐D) seismic refraction data set, demonstrating that tomography is capable of determining a densely sampled velocity model with large velocity contrasts. Forward and inverse modeling procedures are chosen to minimize the computational costs of the inversion. Parameterizing the linearized inversion using functions defined along the ray paths, simple backprojection with zero pixel size is shown to exactly solve the linear problem, producing the smallest model for the slowness perturbation. For small grid cells, simple backprojection closely approximates the exact solution and is a sufficient solution for an iterative nonlinear inversion. This eliminates the need to store or solve a large system of linear equations. Accurate first arrival travel times are rapidly computed using a finite difference algorithm. Forward modeling between each simple backprojection allows the procedure to correctly account for the locations of the rays. This becomes more important as the spatial resolution of the model is improved. The computational efficiency of the entire nonlinear procedure allows the model to be densely sampled, providing a spatially well‐resolved 3‐D tomographic image. The synthetic refraction survey is designed to be similar to a published 3‐D survey over the East Pacific Rise. Tests based on this example and others show that 3‐D tomography is capable of inverting a large travel time data set for detailed earth structure with large lateral velocity variations and is stable in the presence of noisy data.

Inverse scattering, seismic traveltime tomography, and neural networks

AbstractInverse scattering methods for reconstructing sound‐wave‐speed structure in the dimensions have been shown to be equivalent to inverting line integrals when the scattered field is of sufficiently high frequency and the scattering is sufficiently weak. Seismic traveltime tomography uses first arrival traveltime data to invert for wave‐speed structure. Of course, the traveltime is itself a line integral along a refracting ray path through the medium being probed. The similarity between these two inversion problems is discussed. One type of neural network‐the Hopfield net‐may be used to improve the traveltime inversion. We find that, by taking advantage of the general relationship between least‐squares solution and generalized inverses, the neural networks approach eliminates the need for inverting singular or poorly conditioned matrices and therefore also eliminates the need for the damping term often used to regularize such inversions. This procedure produces reconstructions with fewer artifacts and faster convergence than those attained previously using damped least‐squares methods.

Fast Ip solution of large, sparse, linear systems: Application to seismic travel time tomography

Least-squares inversion tends to be sensitive to small changes in the assumptions; in the statistical lexicon, it is not “robust.” Least-squares solutions of noisy, ill-conditioned systems often exhibit unphysical, high-frequency oscillations which have to be damped or smoothed in some fashion. The iteratively reweighted least squares (IRLS) algorithm provides a means of computing approximate l p solutions (1 ⩽ p). We will show that IRLS can be combined with the preconditioned conjugate gradient algorithm in order to solve large, sparse, rectangular systems of linear, algebraic equations very efficiently. Further, for p ≅ 1, the resulting algorithm is extremely stable; it seems to give reasonably smooth solutions without minute adjustment of the parameters and does not become unstable if an overly optimistic error estimate is used. It also has the ability to reject, or at least significantly diminish, the influence of outliers in the data and does not require the use of damping. In addition, we will show how to efficiently include diagonal weighting of the observations and parameters in order to incorporate a priori information into the inversion. This combined inversion algorithm has wide applicability in “real-world” inverse theory, parameter estimation, filtering, etc., where data are noisy and not uniformly reliable. Numerical examples of the l 1 inversion of noisy seismic travel time data will be shown.

Applications of seismic travel-time tomography

Summary. This paper describes the application of tomography to seismic travel-time inversion. There are various implementations of travel-time tomography. In reflection tomography, sources and receivers are on the surface of the Earth and the principal seismic events are reflections from subsurface velocity discontinuities. In transmission tomography, sources and/or receivers may be buried beneath the surface and the events correspond to direct, or unreflected, arrivals; this is the analogue of medical tomography. There are also cases in which both direct as well as reflected arrivals are important, such as in Vertical Seismic Profiling. The latter is a direct application of the first two, but is not discussed in any detail here. It is also shown how the iterative use of travel-time tomography and depth migration can produce much enhanced subsurface images. Examples of both transmission tomography and reflection tomography combined with depth migration illustrate the methods.