Geophysics

Full-waveform inversion (FWI) is a challenging data-fitting procedure based on full-wavefield modeling to extract quantitative information from seismograms. High-resolution imaging at half the propagated wavelength is expected. Recent advances in high-performance computing and multifold/multicomponent wide-aperture and wide-azimuth acquisitions make 3D acoustic FWI feasible today. Key ingredients of FWI are an efficient forward-modeling engine and a local differential approach, in which the gradient and the Hessian operators are efficiently estimated. Local optimization does not, however, prevent convergence of the misfit function toward local minima because of the limited accuracy of the starting model, the lack of low frequencies, the presence of noise, and the approximate modeling of thewave-physics complexity. Different hierarchical multiscale strategies are designed to mitigate the nonlinearity and ill-posedness of FWI by incorporating progressively shorter wavelengths in the parameter space. Synthetic and real-data case studies address reconstructing various parameters, from [Formula: see text] and [Formula: see text] velocities to density, anisotropy, and attenuation. This review attempts to illuminate the state of the art of FWI. Crucial jumps, however, remain necessary to make it as popular as migration techniques. The challenges can be categorized as (1) building accurate starting models with automatic procedures and/or recording low frequencies, (2) defining new minimization criteria to mitigate the sensitivity of FWI to amplitude errors and increasing the robustness of FWI when multiple parameter classes are estimated, and (3) improving computational efficiency by data-compression techniques to make 3D elastic FWI feasible.

I present a finite‐difference method for modeling P-SV wave propagation in heterogeneous media. This is an extension of the method I previously proposed for modeling SH-wave propagation by using velocity and stress in a discrete grid. The two components of the velocity cannot be defined at the same node for a complete staggered grid: the stability condition and the P-wave phase velocity dispersion curve do not depend on the Poisson’s ratio, while the S-wave phase velocity dispersion curve behavior is rather insensitive to the Poisson’s ratio. Therefore, the same code used for elastic media can be used for liquid media, where S-wave velocity goes to zero, and no special treatment is needed for a liquid‐solid interface. Typical physical phenomena arising with P-SV modeling, such as surface waves, are in agreement with analytical results. The weathered‐layer and corner‐edge models show in seismograms the same converted phases obtained by previous authors. This method gives stable results for step discontinuities, as shown for a liquid layer above an elastic half‐space. The head wave preserves the correct amplitude. Finally, the corner‐edge model illustrates a more complex geometry for the liquid‐solid interface. As the Poisson’s ratio v increases from 0.25 to 0.5, the shear converted phases are removed from seismograms and from the time section of the wave field.

The frequency‐dependent properties of Rayleigh‐type surface waves can be utilized for imaging and characterizing the shallow subsurface. Most surface‐wave analysis relies on the accurate calculation of phase velocities for the horizontally traveling fundamental‐mode Rayleigh wave acquired by stepping out a pair of receivers at intervals based on calculated ground roll wavelengths. Interference by coherent source‐generated noise inhibits the reliability of shear‐wave velocities determined through inversion of the whole wave field. Among these nonplanar, nonfundamental‐mode Rayleigh waves (noise) are body waves, scattered and nonsource‐generated surface waves, and higher‐mode surface waves. The degree to which each of these types of noise contaminates the dispersion curve and, ultimately, the inverted shear‐wave velocity profile is dependent on frequency as well as distance from the source. Multichannel recording permits effective identification and isolation of noise according to distinctive trace‐to‐trace coherency in arrival time and amplitude. An added advantage is the speed and redundancy of the measurement process. Decomposition of a multichannel record into a time variable‐frequency format, similar to an uncorrelated Vibroseis record, permits analysis and display of each frequency component in a unique and continuous format. Coherent noise contamination can then be examined and its effects appraised in both frequency and offset space. Separation of frequency components permits real‐time maximization of the S/N ratio during acquisition and subsequent processing steps. Linear separation of each ground roll frequency component allows calculation of phase velocities by simply measuring the linear slope of each frequency component. Breaks in coherent surface‐wave arrivals, observable on the decomposed record, can be compensated for during acquisition and processing. Multichannel recording permits single‐measurement surveying of a broad depth range, high levels of redundancy with a single field configuration, and the ability to adjust the offset, effectively reducing random or nonlinear noise introduced during recording. A multichannel shot gather decomposed into a swept‐frequency record allows the fast generation of an accurate dispersion curve. The accuracy of dispersion curves determined using this method is proven through field comparisons of the inverted shear‐wave velocity ([Formula: see text]) profile with a downhole [Formula: see text] profile.

Magnetotelluric (MT) data are inverted for smooth 2-D models using an extension of the existing 1-D algorithm, Occam’s inversion. Since an MT data set consists of a finite number of imprecise data, an infinity of solutions to the inverse problem exists. Fitting field or synthetic electromagnetic data as closely as possible results in theoretical models with a maximum amount of roughness, or structure. However, by relaxing the misfit criterion only a small amount, models which are maximally smooth may be generated. Smooth models are less likely to result in overinterpretation of the data and reflect the true resolving power of the MT method. The models are composed of a large number of rectangular prisms, each having a constant conductivity. [Formula: see text] information, in the form of boundary locations only or both boundary locations and conductivity, may be included, providing a powerful tool for improving the resolving power of the data. Joint inversion of TE and TM synthetic data generated from known models allows comparison of smooth models with the true structure. In most cases, smoothed versions of the true structure may be recovered in 12–16 iterations. However, resistive features with a size comparable to depth of burial are poorly resolved. Real MT data present problems of non‐Gaussian data errors, the breakdown of the two‐dimensionality assumption and the large number of data in broadband soundings; nevertheless, real data can be inverted using the algorithm.

A gridding method commonly called minimum curvature is widely used in the earth sciences. The method interpolates the data to be gridded with a surface having continuous second derivatives and minimal total squared curvature. The minimum‐curvature surface has an analogy in elastic plate flexure and approximates the shape adopted by a thin plate flexed to pass through the data points. Minimum‐curvature surfaces may have large oscillations and extraneous inflection points which make them unsuitable for gridding in many of the applications where they are commonly used. These extraneous inflection points can be eliminated by adding tension to the elastic‐plate flexure equation. It is straightforward to generalize minimum‐curvature gridding algorithms to include a tension parameter; the same system of equations must be solved in either case and only the relative weights of the coefficients change. Therefore, solutions under tension require no more computational effort than minimum‐curvature solutions, and any algorithm which can solve the minimum‐curvature equations can solve the more general system. We give common geologic examples where minimum‐curvature gridding produces erroneous results but gridding with tension yields a good solution. We also outline how to improve the convergence of an iterative method of solution for the gridding equations.

Từ Định luật Ampere (với một trái đất đồng nhất) và từ phương trình Maxwell sử dụng khái niệm vectơ Hertz (cho một trái đất nhiều tầng), các giải pháp được tìm ra cho các thành phần ngang của trường điện và từ tại bề mặt do dòng điện đất (telluric currents) trong lòng đất. Tỷ lệ của các thành phần ngang này, cùng với pha tương đối của chúng, là chỉ báo về cấu trúc và điện trở suất thực của các lớp dưới mặt đất. Tỷ lệ của một số cặp yếu tố điện từ khác cũng có tính chỉ báo tương tự. Thông thường, một bảng đo quang điện-lừu từ được thể hiện bằng những đường cong điện trở suất biểu kiến và sự khác biệt pha tại một trạm cụ thể, được vẽ dưới dạng hàm của chu kỳ của các thành phần dòng điện đất khác nhau. Các công thức cụ thể được xây dựng cho điện trở suất, độ sâu tới các mặt phân cách, v.v. trong cả bài toán hai và ba lớp. Đối với hai vùng có hình dạng tương tự và điện trở suất tương ứng của chúng chỉ khác nhau bởi một hệ số tuyến tính, các mối quan hệ về pha là giống nhau và các điện trở suất biểu kiến khác nhau bởi cùng một hằng số tỷ lệ mà liên hệ với các điện trở suất thực tương ứng. Nguyên tắc "tính tương tự" này đơn giản hóa đáng kể việc biểu diễn một bộ đường cong chủ, như đã được đưa ra để sử dụng trong việc giải thích địa chất. Ngoài các lợi thế thông thường mang lại bởi việc sử dụng dòng điện đất (không cần các nguồn dòng điện hoặc cáp dài, độ sâu khảo sát lớn hơn, v.v.), phương pháp điện-lừu-từ trong thăm dò địa chất giải quyết các hiệu ứng của từng lớp đất tốt hơn so với các phương pháp điện trở thông thường. Nó dường như là một công cụ lý tưởng để điều tra ban đầu các lưu vực trầm tích lớn có tiềm năng dự trữ dầu mỏ.

The purpose of this paper is to discuss field and interpretive techniques which permit, in favorable cases, the quite accurate determination of seismic interval velocities prior to drilling. A simple but accurate formula is developed for the quick calculation of interval velocities from “average velocities” determined by the known [Formula: see text] technique. To secure accuracy a careful study of multiple reflections is necessary and this is discussed. Although the principal objective in determining velocities is to allow an accurate structural interpretation to be made from seismic reflection data, an important secondary objective is to get some lithological information. This is obtained through a correlation of velocities with rock type and depth.

From experimental studies in digital processing of seismic reflection data, geophysicists know that a seismic signal does vary in amplitude, shape, frequency and phase, versus propagation time. To enhance the resolution of the seismic reflection method, we must investigate these variations in more detail. We present quantitative results of theoretical studies on propagation of plane waves for normal incidence, through perfectly elastic multilayered media. As wavelet shapes, we use zero‐phase cosine wavelets modulated by a Gaussian envelope and the corresponding complex wavelets. A finite set of such wavelets, for an appropriate sampling of the frequency domain, may be taken as the basic wavelets for a Gabor expansion of any signal or trace in a two‐dimensional (2-D) domain (time and frequency). We can then compute the wave propagation using complex functions and thereby obtain quantitative results including energy and phase of the propagating signals. These results appear as complex 2-D functions of time and frequency, i.e., as “instantaneous frequency spectra.” Choosing a constant sampling rate on the logarithmic scale in the frequency domain leads to an appropriate sampling method for phase preservation of the complex signals or traces. For this purpose, we developed a Gabor expansion involving basic wavelets with a constant time duration/mean period ratio. For layered media, as found in sedimentary basins, we can distinguish two main types of series: (1) progressive series, and (2) cyclic or quasi‐cyclic series. The second type is of high interest in hydrocarbon exploration. Progressive series do not involve noticeable distortions of the seismic signal. We studied, therefore, the wave propagation in cyclic series and, first, simple models made up of two components (binary media). Such periodic structures have a spatial period. We present synthetic traces computed in the time domain using the Goupillaud‐Kunetz model of propagation for one‐dimensional (1-D) synthetic seismograms. Three different cases appear for signal scattering, depending upon the value of the ratio wavelength of the signal/spatial period of the medium. (1) Large wavelengths The composite medium is fully transparent, but phase delaying. It acts like an homogeneous medium, with an “effective velocity” and an “effective impedance.” (2) Short wavelengths For wavelengths close to twice the spatial period of the medium, the composite medium strongly attenuates the transmission, and superreflectivity occurs as counterpart. (3) Intermediate wavelengths For intermediate values of the frequency, velocity dispersion versus frequency appears. All these phenomena are studied in the frequency domain, by analytic formulation of the transfer functions of the composite media for transmission and reflection. Such phenomena are similar to Bloch waves in crystal lattices as studied in solid state physics, with only a difference in scale, and we checked their conformity with laboratory measurements. Such models give us an easy way to introduce the use of effective velocities and impedances which are frequency dependent, i.e., complex. They will be helpful for further developments of “complex deconvolution.” The above results can be extended to quasi‐cyclic media made up of a random distribution of double layers. For signal transmission, quasi‐cyclic series act as a high cut filter with possible time delay, velocity dispersion, and “constant Q” type of law for attenuation. For signal reflection they act as a low cut filter, with possible superreflections. These studies could be extended to three‐dimensional (3-D) binary models (grains and pores in a porous reservoir), in agreement with well‐known acoustic properties of gas reservoirs (theory of bright spots). We present some applications to real well data.

The compressional wave reflection coefficient R(θ) given by the Zoeppritz equations is simplified to the following: [Formula: see text] The first term gives the amplitude at normal incidence (θ = 0), the second term characterizes R(θ) at intermediate angles, and the third term describes the approach to critical angle. The coefficient of the second term is that combination of elastic properties which can be determined by analyzing the offset dependence of event amplitude in conventional multichannel reflection data. If the event amplitude is normalized to its value for normal incidence, then the quantity determined is [Formula: see text] [Formula: see text] specifies the normal, gradual decrease of amplitude with offset; its value is constrained well enough that the main information conveyed is [Formula: see text] where [Formula: see text] is the contrast in Poisson’s ratio at the reflecting interface and [Formula: see text] is the amplitude at normal incidence. This simplified formula for R(θ) accounts for all of the relations between R(θ) and elastic properties first described by Koefoed in 1955.

The utility of multispectral remote sensing techniques for discriminating among materials is based on the differences that exist among their spectral properties. As distinct from spectral variations that occur as a consequence of target condition and environmental factors, intrinsic spectral features that appear in the form of bands and slopes in the visible and near infrared (.325 to 2.5 μm) bidirectional reflection spectra of minerals (and, consequently, rocks) are caused by a variety of electronic and vibrational processes. These processes, such as crystal field effects, charge‐transfer, color centers, transitions to the conduction band, and overtone and combination tone vibrational transitions are discussed and illustrated with reference to specific minerals. Spectral data collected from a large selection of minerals are used to generate a “spectral signature” diagram that summarizes the optimum intrinsic information available from the spectra of particulate minerals. The diagram provides a ready reference for the interpretation of visible and near infrared features that typically appear in remotely sensed data. In the visible‐near infrared region, the most commonly observed features in naturally occurring materials are due to the presence of iron in some form or other, or to the presence of water or OH groups.