THE INTERPRETATION OF INDUCED-POLARIZATION SOUNDING CURVES IN THE FREQUENCY DOMAIN *

Two-layer type curves of apparent frequency effect for the Wenner configuration are presented. The formulation is based on the normal definition of frequency effect in terms of resistivities measured at different frequencies plus the definition of apparent resistivity over two horizontal layers as a function of first and second layer resistivities. The use of these type curves in the interpretation of multilayer apparent frequency-effect curves is described and some field examples are given.     

CALCULATION OF SYNTHETIC REFLECTION SEISMOGRAMS IN THE FREQUENCY DOMAIN *

Calculation of reflection seismograms in the frequency domain makes it possible to treat absorption correctly, i.e. makes it possible to include dispersion (= causality) and frequency dependent reflection coefficients. In the paper it is shown how such a calculation can be done in a reasonably efficient manner.     

OPTIMUM DIGITAL FILTERING AND INVERSE FILTERING IN THE FREQUENCY DOMAIN*

Two distinct filters are developed in the frequency domain which represent an attempt to increase the resolution of fine structure contained in the signal whilst keeping the expected filtered noise energy within reasonable bounds. A parameter termed the White Noise Amplification is defined and used together with a measure of the deconvolved pulse width in order to provide a more complete characterisation of the filters. Each of the two main types of frequency domain filters discussed varies in properties with respect to a single adjustable parameter. This may be contrasted with a time domain Wiener filter which in general has three variables: length, delay and an adjustable noise parameter or weight. The direct frequency domain analogue of the Wiener filter is termed a gamma-Fourier filter, and is shown to have properties which span the range from those of a spiking filter with zero least square error at one extreme, to those of a matched filter at the other extreme of its variable parameter's range. The second type of filter considered—termed the modulated Gaussian filter—is similarly shown to be a perfect spiking filter at one extreme of its parameter range, but adopts the properties of an output energy filter at the other extreme.     

ANALYSIS OF LEAST SQUARES SMOOTHING OPERATORS IN THE FREQUENCY DOMAIN*

The process of data smoothing by least squares operators with real and symmetric coefficients is shown to be equivalent to linear digital filtering. Study of the frequency response functions of the least squares operators shows that these operators are essentially low-pass filters. The frequency response functions are characterized by the presence of a main lobe centered about the zero frequency together with a number of minor lobes of smaller amplitudes at higher frequencies. The predominant band of frequencies which is passed by a least squares operator occurs within the main lobe of the frequency response function. The filtering characteristics of a least squares operator are dependent on the degree of the polynomial and the number of data points. Generally speaking, the higher the degree of the polynomial the broader the pass band of the filter will be. But the effect of the number of data points on the pass band is exactly opposite. Several figures showing the plots of the half-power frequency, the quarter-power frequency, and the half-width of the main lobe for a large number of least squares operators with different combinations of the degree of the polynomial and the number of data points are presented in this paper. These figures will be helpful to those people who wish to make a selection of least squares operators suitable for their purposes.     

COMPARATIVE ANALYSIS OF TIME DOMAIN AND FREQUENCY DOMAIN IN THE INDUCED POLARIZATION PROSPECTING METHOD*

The polarization content of a medium, in both the time and frequency domains, can be described by parameters which differ in inherent physical meaning and their practical significance. For real situations, general expressions for the apparent parameters, previously determined for both domains, exist for the general case of soundings on a horizontally multi-layered earth. The comparative analysis of these expressions, here restricted to the simple case of a two-layered earth, shows that the theoretical sounding curves of the frequency-domain are different from those of the time-domain. In particular, for every resistivity or chargeability contrasts examined, the apparent frequency-effect curve lies always over the corresponding apparent chargeability curve, but both curves reach the same asymptotical values for shortest and largest spacings. The important conclusions which can be drawn from this result is that both techniques are suitable to investigate subsoil polarizability anomalies. However, from a practical point of view, it is more convenient to adopt the frequency-domain technique when the polarizability increases with depth, while, on the contrary, the time-domain technique is more efficacious when the polarizability decreases with depth.     

频率域中的合成地震记录

波动理论的合成地震记录是地震资料处理和解释中的重要研究课题。 目前,制作波动理论的合成记录主要是用Trorey的克希霍夫方法,正在研究的有有限差分法和有限元素法。本文提出一种新的方法,在频率域中制作波动理论的合成记录,给出了与克希霍夫绕射波方程等价的频率域中的绕射波方程。在国产DJS-11型计算机上,应用两种绕射波方程实现了合成地震记录,并且进行了比较,认为频率域方法是有效的,可以用于波动方程偏移和地震资料解释,进一步工作有可能获得多层变速模型的波动理论记录。     

频率域中的合成地震记录

波动理论的合成地震记录是地震资料处理和解释中的重要研究课题。 目前,制作波动理论的合成记录主要是用Trorey的克希霍夫方法,正在研究的有有限差分法和有限元素法。本文提出一种新的方法,在频率域中制作波动理论的合成记录,给出了与克希霍夫绕射波方程等价的频率域中的绕射波方程。在国产DJS-11型计算机上,应用两种绕射波方程实现了合成地震记录,并且进行了比较,认为频率域方法是有效的,可以用于波动方程偏移和地震资料解释,进一步工作有可能获得多层变速模型的波动理论记录。     

FORWARD MODELING IN THE FREQUENCY-SPACE DOMAIN*

Forward modeling of zero-offset data is performed in the frequency-space domain using a one-way extrapolation equation. The use of the frequency domain offers several advantages over conventional time domain methods. The greatest advantage of the frequency domain is that all time derivatives are evaluated exactly by a simple multiplication. Synthetic zero-offset sections are computed with a high degree of accuracy for arbitrary velocity and reflectivity structures. Examples are shown for realistic complicated models and compared with results from physical modeling.     

A FREQUENCY DOMAIN APPROACH TO TWO-DIMENSIONAL MIGRATION *

The problem of the propagation of acoustic waves in a two-dimensional layered medium can be easily solved in the frequency domain if the Dix approximation is used, i.e. when only the primary reflections are considered. The migrated data at a depth z are obtained by convolving the time section with a proper two-dimensional operator dependent on z. The same result can be obtained by multiplying their two-dimensional spectra and summing for all the values of the temporal frequency. The aspect of the operator in the time-space domain has the classic hyperbolic structure together with the prescribed temporal and spatial decay. The main advantages of the frequency domain approach consist in the noticeable computer time savings and in the better approximation. On the other hand lateral velocity variations are very difficult to be taken into account. This can be done if a space variant filter is used in the time-space domain. To reduce computer time, this filter has to be recursive; the problem has been solved by Claerbout by transforming the hyperbolic partial differential equation into a parabolic one, and using the latter to generate the recursion operator. In the presentation a method is given for the generation of recursive filters with a better phase characteristics that have a pulse response with the requested hyperbolic shape instead of the parabocli one. This allows a better migration of steeper dips.     

频率域磁性单界面反演方法

作者在广泛研究了前人常J磁性单界面反演方法的基础上,导出了常J与变J单界面反问题解的近似表达式,探讨了解的稳定求法,给出了解的精确化措施。据此,调试成功了反演速度较快、精度较高、抗高频干扰能力较强、应用范围较广的常J与变J单界面反演程序。理论模型和实际资料反演结果证明了方法的可靠性。     

FORWARD MODELLING FOR FREQUENCY DOMAIN MARINE ELECTROMAGNETIC SYSTEMS*

An algorithm for the accurate evaluation of rapidly oscillating integrals is described. The method is based on deformation of the integration path into the complex plane of the integration variable. Numerical integration is then carried out along appropriate cuts where the oscillating factor is transformed to the decaying factor. Contrary to standard methods, the proposed technique permits accurate evaluation of numerically divergent integrals. The algorithm is especially useful in forward modelling for high-frequency electromagnetic methods and, in particular, for the new marine electromagnetic system based on measuring signals on the sea bottom at high induction numbers. Results of calculations using both the proposed and standard methods are compared with available analytical solutions.     

RESIDUAL STATICS ESTIMATION BY STACK-POWER MAXIMIZATION IN THE FREQUENCY DOMAIN1

Traditionally, residual static corrections are based on timeshifts estimated for individual CMP sorted traces, which are later resolved into surface-consistent statics. This is a stable and attractive procedure because the data flow is simple and the memory storage required is limited. An alternative station-oriented method maximizing the stack-power estimates surface-consistent static corrections directly. The statics evaluation in this method involves several CMP gathers, which should improve the prediction of statics on noise-contaminated data. In this paper the performance of the above methods will be compared using synthetic as well as real seismic data. Neither method is capable of estimating large statics compared to the dominating period, because local optimization might fail. Global Monte Carlo search by, for instance, simulated annealing has been used to overcome the cycle-skipping problems when proper field statics are missing. Although this procedure is computationally very heavy, it may be the only way to deal with large residual statics. In order to enlarge the operational field for local optimization, it is suggested that the stack-power in the frequency domain is maximized. This makes it easy to change the frequency band during the optimization. Making use of the frequency domain will also normally be faster than the traditional time-domain optimization even for a limited number of iterations. Moreover, the main memory storage required can be significantly reduced, since it is only necessary to keep the frequency band in the memory, where the signal-to-noise ratio is good.     

频率域磁性单界面反演方法

作者在广泛研究了前人常J磁性单界面反演方法的基础上,导出了常J与变J单界面反问题解的近似表达式,探讨了解的稳定求法,给出了解的精确化措施。据此,调试成功了反演速度较快、精度较高、抗高频干扰能力较强、应用范围较广的常J与变J单界面反演程序。理论模型和实际资料反演结果证明了方法的可靠性。     

声波方程频率域有限元参数反演

推导出频率域有限元声波正演方程,为了消除边界反射,将Clayton-Engquist旁轴波动方程吸收边界条件引入频率域,并对有限元刚度矩阵和质量矩阵进行压缩存储,利用广义共轭梯度法求解有限元方程获得正演解.在此基础上,推导出在某一频率下波场数据残差δU与单元物性参数修改量δλ之间关系的Jacobi矩阵,反演方法允许利用地面二维炮集全波场资料与给出初始模型参数的正演值的差值δU,迭代求得δλ.由于计算机内存的限制,方法计算不允许有过多数目的未知数个数,因此还提出了对同一介质物性单元的Jacobi矩阵元素进行压缩组装的措施,从而使反演的未知量个数减少,结合采用共轭梯度迭代法,使得只需利用有效波频段的少数一些频率即可进行迭代反演.正演和反演理论模型的数值模拟结果表明方法是有效的.     

声波方程频率域有限元参数反演

推导出频率域有限元声波正演方程,为了消除边界反射,将Clayton-Engquist旁轴波动方程吸收边界条件引入频率域,并对有限元刚度矩阵和质量矩阵进行压缩存储,利用广义共轭梯度法求解有限元方程获得正演解.在此基础上,推导出在某一频率下波场数据残差δU与单元物性参数修改量δλ之间关系的Jacobi矩阵,反演方法允许利用地面二维炮集全波场资料与给出初始模型参数的正演值的差值δU,迭代求得δλ.由于计算机内存的限制,方法计算不允许有过多数目的未知数个数,因此还提出了对同一介质物性单元的Jacobi矩阵元素进行压缩组装的措施,从而使反演的未知量个数减少,结合采用共轭梯度迭代法,使得只需利用有效波频段的少数一些频率即可进行迭代反演.正演和反演理论模型的数值模拟结果表明方法是有效的.     

桩基检测中反射波法频域曲线的分析与研究

根据波动理论探讨了反射波在砼榫中的传播原理及频率特性，分析了频域曲线判读中的一 影响因素，并结合工程实例讨论了频域分析法的应用及存在的问题。     

MIGRATION IN TERMS OF SPATIAL DECONVOLUTION*

The relationship between two finite-difference schemes (15° and 40°) and the Kirchhoff summation approach is discussed by using closed form solutions of Claerbout's approximate versions of the wave equation. Forward extrapolation is presented as a spatial convolution procedure for each frequency component. It is shown that downward extrapolation can be considered as a wavelet deconvolution procedure, the spatial wavelet being given by the wave theory. Using this concept, a three-dimensional model for seismic data is proposed. The advantages of downward extrapolation in the space-frequency domain are discussed. Finally, it is derived that spatial sampling imposes an upper limit on the aperture and a lower limit on the extrapolation step.     

USE OF THE SEISMIC DYNAMIC DECONVOLUTION ALGORITHM IN DIRECT RESISTIVITY INTERPRETATION*

Seismic dynamic deconvolution is the mathematical basis on which a degree of unification in different prospecting methods is possible, relative to the parameter identification in horizontally stratified media. There is a basic structure which has some immediate applications to the inversion of resistivity data and possibly to other problems. For resistivity soundings there exists a key equation which is parallel to the energy conservation law in the theory of synthetic seismograms.     

CONSISTENT INFINITESIMAL FINITE-ELEMENT CELL METHOD IN FREQUENCY DOMAIN

To calculate the dynamic-stiffness matrix at the structure–medium interface of an unbounded medium for the range of frequencies of interest, the consistent infinitesimal finite-element cell method based on finite elements is developed. The derivation makes use of similarity and finite-element assemblage, yielding a non-linear first-order ordinary differential equation in frequency. The asymptotic expansion for high frequency yields the boundary condition satisfying the radiation condition. In an application only the structure–medium interface is discretized resulting in a reduction of the spatial dimension by one. The boundary condition on the free surface is satisfied automatically. The consistent infinitesimal finite-element cell method is exact in the radial direction and converges to the exact solution in the finite-element sense in the circumferential directions. Excellent accuracy results.