Calculation of magnetic anomalies caused by 2D bodies of arbitrary shape with consideration of demagnetization

Forward calculations of magnetic anomalies caused by two-dimensional bodies of any shape and magnetic properties may be performed either without considering demagnetization as in the equivalent source technique or taking demagnetization into account as in the volume integral equation (VIE) approach, in which, for this purpose, magnetized bodies are divided into a set of rectangular prismatic cells. Ignoring demagnetization may result in distortion of the shape and the amplitude of an anomaly, whereas rectangular cells may not be an optimal representation of the source. Moreover, an inaccurate form approximation in the VIE technique may lead to inconsistent results in the near-body region. In this paper, a method is proposed, based on the VIE approach but differing by applying triangular elementary cells. The method largely overcomes the above-mentioned limitations of the VIE technique. It allows us to delineate large and complex structures exactly and only requires the source to be divided into a few elementary cells to take demagnetization into account satisfactorily. These improvements have been attained through analytical calculation of the Green's function in the complex plane, using the theory of the Cauchy-type integral. Comparing numerical solutions with analytical solutions for homogeneous elliptic cylinders without remanence, the method is found to be consistent with the theory in the range of relative magnetic permeability of 2–20, not only far from but also at subcell distances from the body. The method is appropriate for modelling highly and inhomogeneously magnetized 2D bodies of any shape. It may be of value in interpreting underground measurements or topographic effects, as well as in modelling regional geomagnetic profiles, and it is also a convenient tool for testing questionable geological hypotheses. In the framework of the method, the gravitational anomaly for the same causative bodies can be easily calculated. However, at higher and geologically uncommon values of relative magnetic permeability, the algorithm may become unstable but may be stabilized with SVD regularization. The fact that discrepancies were found with the method employed is a basis for further research.     

Green's function for three-dimensional elastic wave equation with a moving point source on the free surface with applications

High-speed train seismology has come into being recently. This new kind of seismology uses a high-speed train as a repeatable moving seismic source. Therefore, Green's function for a moving source is needed to make theoretical studies of the high-speed train seismology. Green's function for three-dimensional elastic wave equation with a moving point source on the free surface is derived. It involves a line integral of the Green's function for a fixed point source with different positions and corresponding time delays. We give a rigorous mathematical proof of this Green's function. According to the principle of linear superposition, we have also obtained the Green's function for a group of moving sources which can be regarded as a model of a traveling high-speed train. Based on a temporal convolution, an analytical formula for other moving sources is also given. In terms of a moving Gaussian source, we deal with the issue of numerical calculations of the analytical formula. Applications to modelling of a traveling high-speed train are presented. We have considered both the land case and the bridge case for a traveling high-speed train. The theoretical seismograms show different waveform features for these two cases.     

Improving Kirchhoff migration with repeated local plane-wave imaging? A SAR-inspired signal-processing approach in prestack depth imaging

A local plane-wave approach of generalized diffraction tomography in heterogeneous backgrounds, equivalent to Kirchhoff summation techniques when applied in seismic reflection, is re-programmed to act as repeated synthetic aperture radar (SAR) imaging for seismic prestack depth migration. Spotlight-mode SAR imaging quickly provides good images of the electromagnetic reflectivity of the ground via fast Fourier transform (FFT)-based signal processing. By calculating only the Green's functions connecting the aircraft to the centre of the illuminated patch, scattering structures around that centre are also recovered. SAR technology requires us to examine seismic imaging from the local point of view, where the quantity and quality of the available information at each image point are what are important, regardless of the survey geometry. When adapted to seismics, a local image of arbitrary size and sampling is obtained by FFT of seismic energy maps in the scattering wavenumber domain around each node of a pre-calculated grid of Green's functions. These local images can be used to generate a classic prestack depth-migrated section by collecting only their centres. However, the local images also provide valuable information around the centre, as in SAR. They can therefore help to pre-analyse prestack depth migration efficiently, and to perform velocity analysis at a very low cost. The FFT-based signal-processing approach allows local, efficient and automatic control of anti-aliasing, noise and resolution, including optimized Jacobian weights. Repeated local imaging could also be used to speed up migration, with interpolation between local images associated with a coarse grid of Green's functions, as an alternative to interpolation of Green's functions. The local images may, however, show distortions due to the local plane-wave approximation, and the velocity variations across their frame. Such effects, which are not necessarily a problem in SAR, should be controlled and corrected to further enhance seismic imaging. Applications to realistic models and to real data show that, despite the distortion effects, the local images can yield similar information to prestack depth migration, including common-image-point gathers for velocity analyses and AVO/AVA effects, at a much lower cost when a small target is considered.     

Empirical green's function corrected for source effect

One of the severe problems in the semi-empirical method for the prediction of strong ground motions is that there is no objective criterion for choosing empirical Green's functions. It is undesirable that synthesized strong ground motions are affected by the source process of an earthquake whose record is adopted as an empirical Green's function. Through the analysis of strong motion accelerograms of two aftershocks of the 1983 Japan Sea earthquakes, it is found that characteristics of the accelerograms are dependent on the moment rate function derived from teleseismic observations. A procedure is presented for removing the effect of the source process from observed strong motion accelerograms. The thus obtained empirical Green's function expresses approximately the impulse response of the medium between the earthquake source and the observation site.     

SEISMOGRAM SYNTHESIS AND RECOMPRESSION OF DISPERSIVE IN-SEAM SEISMIC MULTIMODE DATA USING A NORMAL-MODE SUPERPOSITION APPROACH

In spite of a geometrical rotation into radial and transverse parts, two- or three-component in-seam seismic data used for underground fault detection often suffer from the problem of overmoding ‘noise’. Special recompression filters are required to remove this multimode dispersion so that conventional reflection seismic data processing methods, e.g. CMP stacking techniques, can be applied afterwards. A normal-mode superposition approach is used to design such multimode recompression filters. Based on the determination of the Green's function in the far-field, the normal-mode superposition approach is usually used for the computation of synthetic single- and multi-mode (transmission) seismograms for vertically layered media. From the filter theory's point of view these Green's functions can be considered as dispersion filters which are convolved with a source wavelet to produce the synthetic seismograms. Thus, the design of multimode recompression filters can be reduced to a determination of the inverse of the Green's function. Two methods are introduced to derive these inverse filters. The first operates in the frequency domain and is based on the amplitude and phase spectrum of the Green's function. The second starts with the Green's function in the time domain and calculates two-sided recursive filters. To test the performance of the normal-mode superposition approach for in-seam seismic problems, it is first compared and applied to synthetic finite-difference seismograms of the Love-type which include a complete solution of the wave equation. It becomes obvious that in the case of one and two superposing normal modes, the synthetic Love seam-wave seismograms based on the normal-mode superposition approach agree exactly with the finite-difference data if the travel distance exceeds two dominant wavelengths. Similarly, the application of the one- and two-mode recompression filters to the finite-difference data results in an almost perfect reconstruction of the source wavelet already two dominant wavelengths away from the source. Subsequently, based on the dispersion analysis of an in-seam seismic transmission survey, the normal-mode superposition approach is used both to compute one- and multi-mode synthetic seismograms and to apply one- and multimode recompression filters to the field data. The comparison of the one- and two-mode synthetic seismograms with the in-seam seismic transmission data reveals that arrival times, duration and shape of the wavegroups and their relative excitation strengths could well be modelled by the normal-mode superposition approach. The one-mode recompressions of the transmission seismograms result in non-dispersive wavelets whose temporal resolution and signal-to-noise ratio could clearly be improved. The simultaneous two-mode recompressions of the underground transmission data show that, probably due to band-limitation, the dispersion characteristics of the single modes could not be evaluated sufficiently accurately from the field data in the high-frequency range. Additional techniques which overcome the problem of band-limitation by modelling all of the enclosed single-mode dispersion characteristics up to the Nyquist frequency will be mandatory for future multimode applications.     

Optimum strong-motion array geometry for source inversion—II

Optimum strong-motion array geometry for source inversions is again determined for each of three types of earthquake faults: strike-slip, dip-slip and offshore subduction thrust. The method is the same as employed in a previous study;¹ however, use of a complete Green's function in an elastic half-space provides better results for engineering practice. It is found that the complete Green's function is capable of stabilizing the accuracy of an inversion solution obtained using theoretical seismograms, regardless of the differences in array configuration. The optimum strong-motion array for a strike-slip fault is characterized by stations well distributed in azimuth, while the optimum array for a dip-slip event has stations arranged in a grid-shaped form. The array geometries obtained here are grossly similar to those in the previous study,¹ which were derived using only the far-field S waves, and are more consistent with those proposed at the 1978 International Workshop on Strong-Motion Earthquake Instrument Arrays.².     

用改进的经验格林函数方法模拟唐山地震动

建筑物的抗震设防需要尽可能地掌握未来大地震强震动记录信息,但大地震强震动记录的匮乏阻碍了抗震设防实践的发展。经验格林函数方法作为模拟地震动的主要方法,可以提供可靠的大地震强震动记录,但也存在着许多问题,如缺乏对大地震断层滑动分布不均匀的描述、用经验确定小震数目、模拟方法受到大小地震相似条件的限制等。文中对上述经验格林函数方法存在的问题进行了研究,改进的经验格林函数方法,有效地解决了上述问题。并用其对唐山大地震进行了模拟,并把模拟的地震动时程和反应谱与实际记录相比较,发现用改进方法模拟的地震动加速度反应谱比用未改进方法模拟结果更接近实际的地震动记录加速度反应谱。由此说明改进的经验格林函数可更准确的模拟地震动。     

Magnetic field generation by the motion of a highly conducting fluid

Abstract

Using an asymptotic expansion of Green's function for the problem of magnetic field generation by 3D steady flow of highly conducting fluid a general antidynamo theorem is proved in the case of no exponential stretching of liquid particles. Explicit formulae connecting the spectrum of the magnetic modes with the geometry of the Lagrangian trajectories are obtained. The existence of the fast dynamo action for special flows with exponential stretching is proved under the condition of smoothness of the fields of stretching and non-stretching directions.     

谱矩方法在磁源体深度反演中的应用研究

谱矩方法可以对数据的表面形貌做较为细致的描述.它以随机过程为理论基础,用各阶谱矩及统计不变量等具体的参数表征表面的几何形态,算术平均顶点曲率是一种基于四阶谱矩的统计不变量.通常,埋深不同的场源所引起的磁异常尺度不同,从曲率的角度来理解即为磁异常曲面的弯曲程度不同.因此,本文应用算术平均顶点曲率提取磁异常的几何信息,并将所提取的信息用于场源深度的反演.理论上推导了基于谱矩的球状磁源体和板状磁源体的反演公式,得到了场源深度与磁异常、曲率之间的关系式.结合理论模型计算验证了方法的有效性,并与欧拉反褶积方法进行对比.与传统的方法相比,该方法快速简单,无需调节参数,且有较好的反演精度.最后,将该方法用于塔里木盆地航磁异常的反演和解释中,反演出的磁源体深度可满足区域磁异常数据分析和解释的要求,为克拉通沉积盆地磁异常源的深度划分提供丰富的信息.     

重力位谱分析及重力异常导数换算新方法--余弦变换

为了提高重力异常导数换算的精度,真实有效地反映地质体的异常特征,提出用余弦变换计算异常导数的新方法. 给出并证明了两个定理,利用它们推导出重力位余弦谱一般表达式以及重力异常各阶导数计算公式,建立了位场余弦谱分析理论. 模型实验中发现,用Fourier变换计算的一阶导数与理论导数偏差很大,而余弦变换计算的导数与理论异常导数拟合效果非常好,除边界几个数据因重力异常的有限截断产生的吉布斯效应残留使误差较大外,数据的计算精度均很高,误差为-009%～5%.     

利用Hilbert变换计算重力归一化总梯度

针对提高重力勘探正反演解释的分辨率问题,提出利用Hilbert变换计算和研究重力归一化总梯度.文中从理论上详细地证明了方法的可行性,给出了适合计算机实现的计算方法.为探讨该方法对油气藏的分辨能力,在模型计算中,分别利用Fourier级数法、Fourier变换法和本文提出的Hilbert变换法计算模型的GH场值,发现当三度体储油球冠模型（模拟似三度贮油气藏背斜模型）油气藏部分（低密度体）厚度减小到低于球冠厚度的十分之一时,Fourier级数及变换法不能分辨出低密度体所产生的异常,而Hilbert变换法仍然可以清晰地识别,这说明用Hilbert变换法计算的GH场对异常识别的分辨率优于其他两种方法.     

利用余弦变换计算重力异常的向上延拓

利用余弦变换计算重力异常的向上延拓是一种新方法.根据余弦变换的基本性质,推导了二度、三度体异常向上延拓余弦变换谱理论公式,采用离散余弦变换实现了该法的数值计算;研究了无限长水平圆柱体的补偿因子中主频段的特性,给出了二度体的线性补偿方式;补偿后的理论模型异常向上延拓具有较高的计算精度,除边部几个数据因数据的离散和有限截断使误差较大外（最大误差为6.23%）,其余数据的误差均在1%以内,理论值和计算值曲线基本重合.这说明,与Fourier变换相比,离散余弦变换在数值计算中,受非周期性深度因子的影响小,补偿方式易于选择,其计算方法优于Fourier变换.     

Analysis of a vertical dike,infinitely deep,striking north by fourier transform

Summary The present work deals with the analysis of the frequency spectrum of a vertical magnetic dike like body, infinitely deep, striking north to determine its width and depth from the surface. Fourier transform of the functional representative of the theoretical magnetic anomaly for such a body has been obtained. By framing amplitude response curves, the parameters were estimated.     

ANALYSIS OF GRAVITY ANOMALIES OF TWO-DIMENSIONAL FAULTS USING FOURIER TRANSFORMS*

The Fourier transform formula for a two-dimensional fault truncating a horizontal bed at an arbitrary angle of inclination is derived. The amplitude spectrum of the Fourier transform is found to give information about the depth to the top of the upper part of the faulted bed and the inclination of the fault-plane. Under suitable conditions the thickness and the displacement of the bed involved can be obtained. With actual field data, these transforms can be obtained at discrete points by a Fourier analysis of the gravity anomaly. A field example from the Logan fault area near Montreal, Que., Canada, is given.     

Source Time Function of Seismic Events at Rudna Copper Mine, Poland

—?The empirical Green's function deconvolution technique is used to retrieve the source time functions from the records of P waves of seven seismic events that occurred at the Rudna copper mine in 1996 and were located in the middle of the underground network. Their moment magnitudes ranged from 2.1 to 2.9. The records of smaller events from the same area and with similar source mechanism, with moment magnitudes ranging from 1.5 to 2.0, were accepted as empirical Green's functions. The relative source time functions were successfully retrieved at a number of stations for six events. Directivity effects, implying unilateral rupture propagation, were observed in five cases. The azimuth of rupture propagation direction and the rupture velocity were estimated from the distribution of pulse widths and pulse maximum amplitudes as a function of the cosine of station azimuths. The rupture propagated approximately either from south to north or from north to south. The rupture velocity was low, ranging from 0.25 to 0.54 of the shear-wave velocity. The source dimensions, represented by the fault length, were also small in comparison with those estimated in the frequency domain and ranged from 80 to 250?m.     

Consistency of the spatial autocorrelation method with seismic interferometry and its consequence

We have cross‐checked the conventional theory of the spatial autocorrelation method and the consequence of seismic interferometry: the retrieval of the elastodynamic Green's function. Their mutual consistency is almost complete. The basic formulas of the conventional spatial autocorrelation theory can be derived by an alternative approach based on the retrieval of the elastodynamic Green's function. The only discrepancy is found with the average of the complex coherence function over azimuth in a wavefield dependent on azimuth. It is hypothesized, in discussion, that this discrepancy is due to the way of representing the wavefield in the background theory of seismic interferometry that can produce only wavefields moderately dependent on azimuth and that the mentioned consequence of seismic interferometry can also only make sense in a wavefield moderately dependent on azimuth. Our field experiment with a wavefield dependent on azimuth showed that the consequence of seismic interferometry in the logical framework of the conventional spatial autocorrelation theory is appropriate under such degrees of approximation as the measure proposed in this study, i.e., the deviation of the total dispersion curves is between about 10 and 16 per cent at the maximum from those averaged over azimuth. The acceptance of the retrieval of Green's function gives a proper physical meaning to the complex coherence function: the real part of the elastodynamic Green's function normalized by its zero‐offset version. This makes it possible to take a deterministic approach rather than the statistical one on which the conventional spatial autocorrelation method is based and gives fruitful new aspects and perspectives. For example, the formula for the multi‐mode case is given and the possibility of exploration of two or three dimensional velocity structures is suggested.     

A NEW APPROACH TO VIBROSEIS DECONVOLUTION*

A method is proposed to obviate the shortcomings of conventional deconvolution approaches applied to vibroseis data. The vibroseis wavelet reduces the time domain resolution of the earth's impulse response by restricting its passband. The spectrum of the wavelet is assumed to be a “low quefrency”phenomenon, and hence it can be estimated by low cut cepstral filtering. The wavelet's amplitude spectrum can then be removed by spectral division. By using an approach which is consistent with the principle of maximum entropy, the undetermined portions of the seismogram's Fourier transform can be filled in by autoregressive prediction. The process of initially deconvolving in a restricted passband reduces the enhancement of noise contaminated parts of the spectrum, and the spectral extension scheme increases the time domain resolution of the process.     

Three-dimensional Green's functions for transversely isotropic thermoporoelastic bimaterials

Green's functions for transversely isotropic thermoporoelastic bimaterials are established in the paper. We first express the compact general solutions of transversely isotropic thermoporoelastic material in terms of harmonic functions and introduce eight new harmonic functions. The three-dimensional Green's function having a concentrated liquid source or a concentrated heat source in steady state is completely solved using these new harmonic functions. The analytical results show some new phenomena of the pore fluid pressure increment, thermal increment and stress distributions at the interface. In the two materials, the pore fluid pressure has the same distribution because of the common fluid permeability, but the situation is different for the thermal increment. Shear failure is most likely at the two sources due to the highly degenerated direction of shear stress contours.     

Interpretation of magnetic anomalies by horizontal and vertical derivatives of the analytic signal

Magnetic anomalies are often disturbed by the magnetization direction, so we can’t directly use the original magnetic anomaly to estimate the exact location and geometry of the source. The 2D analytic signal is insensitive to magnetization direction. In this paper, we present an automatic method based on the analytic signal horizontal and vertical derivatives to interpret the magnetic anomaly. We derive a linear equation using the analytic signal properties and we obtain the 2D magnetic body location parameters without giving a priori information. Then we compute the source structural index (expressing the geometry) by the estimated location parameters. The proposed method is demonstrated on synthetic magnetic anomalies with noise. For different models, the proposed technique can both successfully estimate the location parameters and the structural index of the sources and is insensitive to noise. Lastly, we apply it to real magnetic anomalies from China and obtain the distribution of unexploited iron ore. The inversion results are consistent with the parameters of known ore bodies.     

Frequency-domain double-plane-wave least-squares reverse time migration

Least-squares reverse time migration is often formulated as an iterative updating process, where estimating the gradient of the misfit function is necessary. Traditional time domain shot-profile least-squares reverse time migration is computationally expensive because computing the gradient involves solving the two-way wave equation several times in every iteration. To reduce the computational cost of least-squares reverse time migration, we propose a double-plane-wave least-squares reverse time migration method based on a misfit function for frequency-domain double-plane-wave data. In double-plane-wave least-squares reverse time migration, the gradient is computed by multiplying frequency-domain plane-wave Green's functions with the corresponding double-plane-wave data residual. Because the number of plane-wave Green's functions used for migration is relatively small, they can be pre-computed and stored in a computer's discs or memory. We can use the pre-computed plane-wave Green's functions to obtain the gradient without solving the two-way wave equation in each iteration. Therefore, the migration efficiency is significantly improved. In addition, we study the effects of using sparse frequency sampling and sparse plane-wave sampling on the proposed method. We can achieve images with correct reflector amplitudes and reasonable resolution using relatively sparse frequency sampling and plane-wave sampling, which are larger than that determined by the Nyquist theorem. The well-known wrap-around artefacts and linear artefacts generated due to under-sampling frequency and plane wave can be suppressed during iterations in cases where the sampling rates are not excessively large. Moreover, implementing the proposed method with sparse frequency sampling and sparse plane-wave sampling further improves the computational efficiency. We test the proposed double-plane-wave least-squares reverse time migration on synthetic models to show the practicality of the method.