RESIDUAL STATICS BY CDP-LOCALIZED STACK OPTIMIZATION1

Many conventional schemes of automated residual statics estimate time lags between prestack traces (sorted as CDP gathers) and a model section, and transform the collected lags into surface-consistent residuals. The method discussed in this paper aims at improving the lag estimator. ‘Externally generated’ reference traces are avoided and a principle of localized stack optimization is introduced whereby application of a multichannel filter to the stack and evaluation of the normalized power of every filtered trace gives a measure of the stack quality. One may consider the power as a function of all variable (inconsistent) shifts applied to the prestack traces. To obtain a set of optimal lag estimates the power function must be maximized. This power function is complex and the number of its variables prohibits a straightforward search for the maximum. Thus an iterative method must be employed, and steepest descent schemes have proved the most satisfactory. In the actual calculation, the repeated evaluation of the objective can be replaced by the computation of certain cross-correlations. At the last iteration (after five to ten coordinate sweeps), the global behaviour of this correlation gives some indications of how well a prestack trace is adapted to the filtered stack. This information is used to compute a weighting factor to be stored with the lag estimate. At this stage simple statistical procedures are run to eliminate the most unlikely estimates. The remaining ones are transformed to surface-consistent residuals by means of a weighted least-squares inversion according to a model which takes into account the fact that the lags have no fixed reference datum. The efficiency of the method is demonstrated by a field data example into which synthetic anomalies were introduced, and the effect of the new process is compared with that of a ‘classical’ production program using field data with genuine static problems.     

三种算法联合迭代反演求取最佳剩余静校正量

估算剩余静校正量可看作是以激发点及接收点剩余静校正量为模型参数的、以叠加能量为目标函数的全局最优值的搜索问题．当地震记录的剩余静校正量大而信噪比较低时,应用相关统计法不能有效地拾取静校正量．本文分别采用了ＣＭＰ道集内道间互相关法、图像边缘提取的Ｌａｐｌａｃｉａｎ算法及ＤＦＰ算法,三种不同算法联合迭代反演可求取最佳的模型参数,使目标函数值达到或接近于全局最优．理论试算及实际资料处理效果表明,这种方法能够解决低信噪比及大静校正量造成的周期跳跃等静校正问题．     

三种算法联合迭代反演求取最佳剩余静校正量

估算剩余静校正量可看作是以激发点及接收点剩余静校正量为模型参数的、以叠加能量为目标函数的全局最优值的搜索问题．当地震记录的剩余静校正量大而信噪比较低时，应用相关统计法不能有效地拾取静校正量．本文分别采用了ＣＭＰ道集内道间互相关法、图像边缘提取的Ｌａｐｌａｃｉａｎ算法及ＤＦＰ算法，三种不同算法联合迭代反演可求取最佳的模型参数，使目标函数值达到或接近于全局最优．理论试算及实际资料处理效果表明，这种方法能够解决低信噪比及大静校正量造成的周期跳跃等静校正问题．     

RESIDUAL STATICS ESTIMATION: SCALING TEMPERATURE SCHEDULES USING SIMULATED ANNEALING1

Linearized residual statics estimation will often fail when large static corrections are needed. Cycle skipping may easily occur and the consequence may be that the solution is trapped in a local maximum of the stack-power function. In order to find the global solution, Monte Carlo optimization in terms of simulated annealing has been applied in the stack-power maximization technique. However, a major problem when using simulated annealing is to determine a critical parameter known as the temperature. An efficient solution to this difficulty was provided by Nulton and Salamon (1988) and Andresen et al. (1988), who used statistical information about the problem, acquired during the optimization itself, to compute near optimal annealing schedules. Although theoretically solved, the problem of finding the Nulton–Salamon temperature schedule often referred to as the schedule at constant thermodynamic speed, may itself be computationally heavy. Many extra iterations are needed to establish the schedule. For an important geophysical inverse problem, the residual statics problem of reflection seismology, we suggest a strategy to avoid the many extra iterations. Based on an analysis of a few residual statics problems we compute approximations to Nulton–Salamon schedules for almost arbitrary residual statics problems. The performance of the approximated schedules is evaluated on synthetic and real data.     

SELF-MATCHING DECONVOLUTION IN THE FREQUENCY DOMAIN *

In a previous paper the author showed how, by computing an inverse filter in the frequency domain, an automatic compromise could be made between the conflicting requirements to spike a wavelet and to keep the attendant noise amplification within bounds. This paper extends the technique to take account of errors in the estimated shape of the wavelet defined to the deconvolution process. The drastic effects which such errors can have if they are ignored are demonstrated. A novel form of filter–called the “self-matching filter”–is defined which allows the user to limit not only the noise amplification but also the sensitivity of the filter to random uncertainties in the estimated wavelet. This is achieved by whitening the spectrum only within automatically selected pass bands whilst suppressing other noise-dominated or uncertainly defined frequency components. Conventional Wiener filtering is shown to be a special case of this more general filter, namely one in which the wavelet uncertainty is completely ignored. The type of phase spectrum which the output pulse should be designed to possess (e.g. zero phase or minimum phase) is briefly discussed.     

ESTIMATION OF ELASTIC PARAMETERS FROM AVO EFFECTS IN THE TAU-P DOMAIN1

A quantitative AVO algorithm suitable for media with slow lateral parameter variations is developed. The method is based on a target-oriented inversion scheme for estimation of elastic parameters in a locally horizontally stratified medium. The algorithm uses band-limited PP reflection coefficients in the τ-p domain to estimate P- and S-wave velocities, densities and layer thicknesses. To obtain these reflection coefficients, a pre-processing involving the Radon transform and multiple attenuation is necessary. Furthermore, a macromodel for the velocities above the target zone must be found prior to the inversion. Various inversion tests involving synthetic data with white Gaussian noise and modelling errors that are likely to occur in conjunction with real data have been performed. In general, the inversion algorithm is fairly robust, since it is able to reproduce the main features of the reference model: main interface locations and relative contrasts in the three unknown layer parameters are recovered. From a test combining the effect of source directivity, one thin layer and 20% white Gaussian noise, it was found that neglect of the source directivity in the inversion caused the largest errors in the estimates. This indicates that it is very important either to eliminate the source directivity in a preprocessing step, or to take the directivity into account in the present algorithm. Despite these problems it was concluded that the inversion algorithm was able to reproduce the main features of the reference model.     

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。     

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.     

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。     

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。     

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.     

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.     

SEISMIC INVERSION BY A RMS BORN APPROXIMATION IN THE SPACE-TIME DOMAIN1

A new approximate method to calculate the space-time acoustic wave motion generated by an impulsive point source in a horizontally layered configuration is presented. The configuration consists of a stack of fluid layers between two acoustic half-spaces where the source and the receiver are located in the upper half-space. A distorted-wave Born approximation is introduced; the important feature of the method is the assumption of a background medium with vertical varying root-mean-square acoustic wave speed. A closed-form expression for the scattered field in space and time as a function of the contrast parameters is deduced. The result agrees closely with rigorously calculated synthetic seismograms. In the inverse scheme the wave speed and mass density can be reconstructed within a single trace. Results of the inversion scheme applied to synthetic data are shown.     

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。     

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.     

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.