Inversion of potential field data using the structural index as weighting function rate decay

Nonparametric inverse methods provide a general framework for solving potential‐field problems. The use of weighted norms leads to a general regularization problem of Tikhonov form. We present an alternative procedure to estimate the source susceptibility distribution from potential field measurements exploiting inversion methods by means of a flexible depth‐weighting function in the Tikhonov formulation. Our approach improves the formulation proposed by Li and Oldenburg (1996, 1998) , differing significantly in the definition of the depth‐weighting function. In our formalism the depth weighting function is associated not to the field decay of a single block (which can be representative of just a part of the source) but to the field decay of the whole source, thus implying that the data inversion is independent on the cell shape. So, in our procedure, the depth‐weighting function is not given with a fixed exponent but with the structural index N of the source as the exponent. Differently than previous methods, our choice gives a substantial objectivity to the form of the depth‐weighting function and to the consequent solutions. The allowed values for the exponent of the depth‐weighting function depend on the range of N for sources: 0 ≤N≤ 3 (magnetic case). The analysis regarding the cases of simple sources such as dipoles, dipole lines, dykes or contacts, validate our hypothesis. The study of a complex synthetic case also proves that the depth‐weighting decay cannot be necessarily assumed as equal to 3. Moreover it should not be kept constant for multi‐source models but should instead depend on the structural indices of the different sources. In this way we are able to successfully invert the magnetic data of the Vulture area, Southern Italy. An original aspect of the proposed inversion scheme is that it brings an explicit link between two widely used types of interpretation methods, namely those assuming homogeneous fields, such as Euler deconvolution or depth from extreme points transformation and the inversion under the Tikhonov‐form including a depth‐weighting function. The availability of further constraints, from drillings or known geology, will definitely improve the quality of the solution.     

地震危险性分析中专家意见的评定

本文发展了一种加权平均方法，用于综合考虑地震危险性分析中的专家意见。其中假定接近综合结果的专家意见具有的权系数较大，这样可以合理反映专家意见的一致性与差异性，另外，利用随机加权方法估计综合结果及权系数的不确定性，给出其统计特征量。     

重力数据三维共轭梯度聚焦反演及应用

基于共轭梯度算法对欠定线性目标函数进行求解。 为改善目标函数的多解性、 消除多余构造信息影响, 引入粗糙度系数矩阵; 为克服“趋肤效应”, 更好地反映地质体的真实形态, 在模型目标函数中引入深度加权函数; 为更好的反映地质体的某些尖锐构造和边界, 本文对目标函数添加了基于最小支撑泛函的聚焦反演约束。 通过对多种模型的计算, 验证了该方法具有较好的有效性和稳定性, 并将该方法应用于实际重力资料地下密度反演中去, 得到了较好的反演结果。     

Assessing radial transmissivity variation in heterogeneous aquifers using analytical techniques

Pumping tests are one of the most commonly used in situ testing techniques for assessing aquifer hydraulic properties. Numerous researches have been conducted to predict the effects of aquifer heterogeneity on the groundwater levels during pumping tests. The objectives of the present work were as follows: (1) to predict drawdown conditions and to estimate aquifer properties during pumping tests undertaken in radially symmetric heterogeneous aquifers, and (2) to identify a method for assessing the transmissivity field along the radial coordinate in radially symmetric and fully heterogeneous transmissivity fields. The first objective was achieved by expanding an existing analytical drawdown formulation that was valid for a radially symmetric confined aquifer with two concentric zones around the pumping well to an N concentric zone confined aquifer having a constant transmissivity value within each zone. The formulation was evaluated for aquifers with three and four concentric zones to assess the effects of the transmissivity field on the drawdown conditions. The specific conditions under which aquifer properties could be identified using traditional methods of analysis were also evaluated. The second objective was achieved by implementing the inverse solution algorithm (ISA), which was developed for petroleum reservoirs to groundwater aquifer settings. The results showed that the drawdown values are influenced by a volumetric integral of a weighting function and the transmissivity field within the cone of depression. The weighting function migrates in tandem with the expanding cone of depression. The ability of the ISA to predict radially symmetric and log‐normally distributed transmissivity fields was assessed against analytical and numerical benchmarks. The results of this investigation indicated that the ISA method is a viable technique for evaluating the radial transmissivity variations of heterogeneous aquifer settings. Copyright © 2013 John Wiley & Sons, Ltd.     

Genetic algorithm based LQR vibration wireless control of laminated plate using photostrictive actuators

A photostrictive type of opto-electromechanical actuator activated by high-energy lights can introduce actuation and control effects without hard-wired connections.This paper addresses the controllability aspect in wireless vibration control of plate structures via photostrictive actuators.A modal force index,which has taken into account the mode number,the spatial distribution,and the dimension of the actuator,is chosen as an objective function to determine the optimal locations of photostrictive actuators.A linear methodology is proposed in this paper and the vibration equation is written in the standard state-space form.A binary-coded GA based combined optimal placement and LQR(linear quadratic regulator) control scheme has been incorporated,which maximizes the modal force index,the closed loop damping and minimizes input light intensity to the actuators.In the present method only three weighting factors have been used to search optimal Q and R matrices using GA,which reduces chromosome length and hence minimizes computational time.Numerical results demonstrate that the use of strategically positioned actuator patches can effectively control the fundamental modes that dominate the structural vibration.     

Least squares and integral methods for the spherical harmonic analysis of theSq-field

Spherical harmonic coefficients (SHCs) for the daily magnetic variation fields (solar and lunar) and the main field of the earth are usually estimated by the method of least squares applied to a truncated spherical harmonic series. In this paper, an integral method for computing the SHCs for the solar quiet daily magnetic variation fieldSq is described and applied toSq data for May and June 1965. TheSq SHCs thus derived are then compared with the results obtained using both unweighted and weighted versions of the least squares method. The weighting used tends to orthogonalize the least squares terms. The integral and weighted least squares results agree closely for terms up to order 4 and degree 30, but both disagree considerably for the higher degree terms with the results of the unweighted least squares. Errors introduced by the numerical integration can be shown to be small, hence the disagreement between integral and unweighted least squares coefficient sets arises from improper weighting. Also, it is concluded that discrepancies between the geomagnetic northward and eastward component-derived coefficient sets arise from either time-dependent external sources that produce non-local-time, based fields or nonpotential sources and not from truncation of the spherical harmonic series as has previously been suggested.Deceased.     

CALCULATION OF THE VERTICAL GRADIENT OF THE GRAVITY FIELD USING THE FOURIER TRANSFORM*

Fourier transform techniques have been used to calculate the theoretical filter (amplitude) response function of Nth order vertical derivative continuation operation. The amplitude response functions of the vertical gradient and its continuation follow from the same. These response functions are subsequently used to calculate the weighting coefficients suitable for two dimensional equispaced data. A shortening operator has been incorporated to limit the extent of the operator. For comparative study, some of the developed coefficient sets and the one presented in this paper are analysed in the frequency domain and their merits and demerits are discussed.     

ACOUSTICAL IMAGING OF SOURCE RECEIVER COINCIDENT PROFILES*

The first part of this paper examines a special case of acoustical imaging in which the source and the receiver coincide. The benefits of weighting and muting are studied in detail by means of computer modeling. The test model consists of a single planar interface z=z₁, abruptly terminated at x= o. The amplitude and phase responses are computed in the plane z=z₀= o for two separations of neighboring stations, Δx=λ/10 and Δx=λ/2. Six different weighting factors are used in the test. However, in this source-receiver coincident case, three of the weighting factors produce identical responses, so that all six test factors may be represented by only four curves. It is found that when the spatial sampling at the aperture approaches the condition of critical sampling, i.e. Δx=λ/2, only the weighting factor which implicitly takes into account beam steering along the specular reflection path is acceptable. This factor alone keeps the amplitude and the phase curves undistorted until the difference 2 ·ΔR between two neighboring paths reaches approximately λ/2. If we set 2 ·ΔR=λ/2, we may construct a set of curves which we may call quite appropriately muting curves. These curves are physically interpretable only for station separation Δx > λ/4. The muting curves are symmetrical about the line x= 0 and their angular opening depends on spatial separation Δx, depth z, and wavelength λ (which may vary with depth). The second part of this paper suggests how the weighting factor with implicit beam steering can be applied to reconstruction of two and three-dimensional wavefields. Seismic migration of common depth point (CDP) stacked line data is also discussed. This is a hybrid case which presents certain theoretical difficulties. We shall also mention the velocity problem which is inherent to migration of CDP stacked data. The third and final part concerns implementation of the migration of CDP stacked data. When the spatial sampling is between λ/4 and λ/2, the migration process will benefit from beam steering and from muting. The benefits are more subtle when the separation of the traces is less than λ/4. However, in that case the cost of data collection is considerable and often prohibitive. In either case the migration of seismic data can be expedited by use of precalculated tables of migration velocities, ray path distances, and weights (including muting).     

Spatial weighting functions: transient hydraulic tests and heterogeneous media

To improve understanding of property measurements in heterogeneous media, an energy-based weighting function concept is developed. In (assumed) homogeneous media, the instrument spatial weighting function (ISWF) depends only on the energy dissipation distribution set up by the measurement procedure and it reduces to simply inverse sample volume (uniform weighting) for 1-D parallel flow case (ideal permeameter). For 1-D transient flow in homogeneous media, such as with slug tests, the ISWF varies with position and time, with 95% of the total weighting contained within 115 well radii, even late in the test. In the heterogeneous case, the determination of the ISWF is connected to the problem of determining an equivalent hydraulic conductivity (K), where the criterion for equivalence is based on equal energy dissipation rate rather than equal volume discharge. The discharge-based equivalent K (K(E)) and the energy-based equivalent K in heterogeneous media (K(eh)) are not equal in general, with K(eh) typically above the nodal arithmetic mean K. The possibly more fundamental problem is that as one makes K measurements in heterogeneous media at different locations or on different cores of heterogeneous materials, the ISWF will be heterogeneity dependent, implying that the averaging process resulting in the equivalent K value also varies with position. If the testing procedure is transient, then the averaging process varies with time. This suggests a fundamental ambiguity in the interpretation of hydraulic conductivity measurements in heterogeneous media that may impact how we approach modeling and prediction in a practical sense (Molz 2003). Further research is suggested.     

强震记录仪器响应失真校正的权函数方法—时域离散序列数值分析方法与应用研究之二

简要评述了现有强震记录仪器响应失真校正方法的优点和存在的不足。在文献[1]的基础上，提出了对强震记录进行仪器响应失真校正的权函数方法，并推导建立了相应的计算公式。通过设计真实地震动为已知的算例，对加速度摆和速度摆强震仪的未校正记录，和用不同方法获得的校正记录的精度（或误差）进行了对比分析。理论和算例分析表明，本文方法简单、实用，具有很高的计算机精度且无稳定性问题。     

含噪声数据反演的概率描述

根据贝叶斯理论给出了对含噪声地球物理数据处理的具体流程和方法,主要包括似然函数估计和后验概率计算.我们将数据向量的概念扩展为数据向量的集合,通过引入数据空间内的信赖度,把数据噪声转移到模型空间的概率密度函数上,即获得了反映数据本身的不确定性的似然函数.该方法由于避免了处理阶段数据空间内的人工干预,因而可以保证模型空间中的概率密度单纯反映数据噪声,具有信息保真度高、保留可行解的优点.为了得到加入先验信息的后验分布,本文提出了使用加权矩阵的概率分析法,该方法在模型空间直接引入地质信息,对噪声引起的反演多解性有很强的约束效果.整个处理流程均以大地电磁反演为例进行了展示.     

Streamflow estimation using optimal regional dependency function

The determination of spatial dependency of regionalized variable (ReV) is important in engineering studies. Regional dependency function that leads to calculation of weighting coefficients is required in order to make regional or point‐wise estimations. After obtaining this dependency function, it is possible to complete missing records in the time series and locate new measurement station. Also determination of regional dependency function is also useful to understand the regional variation of ReV. Point Cumulative Semi‐Variogram (PCSV) is another methodology to understand the regional dependency of ReV related to the magnitude and the location. However, this methodology is not useful to determine the weighting coefficient, which is required to make regional and point‐wise estimations. However, in Point Semi‐Variogram (PSV) proposed here, weighting coefficient depends on both magnitude and location. Although the regional dependency function has a fluctuating structure in PSV approach, this function gradually increases with distance in PCSV. The study area is selected in Mississippi river basin with 38 streamflow stations used for PCSV application before. It is aimed to compare two different geostatistical models for the same data set. PSV method has an ability to determine the value of variable along with optimum number of neighbour stations and influence radius. PSV and slope PSV approaches are compared with the PCSV. It was shown that slope slope point semi‐variogram (SPSV) approaches had relative error below 5%, and PSV and PCSV methods revealed relative errors below 10%. Copyright © 2009 John Wiley & Sons, Ltd.     

Splatting算法在三维CT迭代算法中的应用

采用代数重建技术重建三维物体,提出了一种基于splatting原理的ART权值计算方法,以增强重建图像的空间分辨率.该算法基于三维锥束扫描方式,采用双三次样条局部基函数作为插值核,并利用其足印函数对物体进行插值来计算像素的权值.与常量插值相比,这种方法对物体的灰度值有更精确的近似,在相同的实验条件下可重建出分辨率高的物体.给出了适用于三维锥束重建的splatting算法的具体实现,并对实验数据进行了实物重建及误差分析.     

半径-斜率域加权反假频地震数据重建

为减小地震数据缺失给地震后续处理工作带来的影响,需要对地震数据进行插值重建.针对反假频插值重建这个难点问题,进行了相关研究,并由此提出了一种改进的R-P（半径-斜率）域加权反假频地震数据插值重建方法.该方法将F-K（频率-波数）谱变换到R-P域,在R-P域设计一个权函数并将其作用于每次的迭代插值过程.通过模型数据和实际数据的测试,证明了该方法具有较好的反假频插值重建能力.     

Point-to-point connectivity,an abstract concept or a key issue for risk assessment studies?

Connectivity of high/low-permeability areas has been recognized to significantly impact groundwater flow and solute transport. The task of defining a rigorous quantitative measure of connectivity for continuous variables has failed so far, and thus there exist a suite of connectivity indicators which are dependent on the specific hydrodynamic processes and the interpretation method. Amongst the many existing indicators, we concentrate on those characterizing connectivity between the points involved in a hydraulic or tracer test. The flow connectivity indicator used here is based on the time elapsed for hydraulic response in a pumping test (e.g., the storage coefficient estimated by the Cooper–Jacob method, S_est). Regarding transport, we select the estimated porosity from the breakthrough curve (_est). According to Knudby and Carrera [Knudby C, Carrera J. On the relationship between indicators of geostatistical, flow and transport connectivity. Adv Water Resour 2005;28(4):405–21] these two indicators measure connectivity differently, and are poorly correlated. Here, we use perturbation theory to analytically investigate the intrinsic relationship between S_est and _est. We find that _est can be expressed as a weighted line integral along the particle trajectory involving two parameters: the transmissivity point values, T, and the estimated values of S_est along the particle path. The weighting function is linear with the distance from the pumping well, thus the influence of the weighting function is maximum at the injection area, whereas the hydraulic information close to the pumping well becomes redundant (null weight). The relative importance of these two factors is explored using numerical simulations in a given synthetic aquifer and tested against intermediate-scale laboratory tracer experiments. We conclude that the degree of connectivity between two points of an aquifer (point-to-point connectivity) is a key issue for risk assessment studies aimed at predicting the travel time of a potential contaminant.     

Three‐dimensional potential field data inversion with L0 quasinorm sparse constraints

The quantitative explanation of the potential field data of three‐dimensional geological structures remains one of the most challenging issues in modern geophysical inversion. Obtaining a stable solution that can simultaneously resolve complicated geological structures is a critical inverse problem in the geophysics field. I have developed a new method for determining a three‐dimensional petrophysical property distribution, which produces a corresponding potential field anomaly. In contrast with the tradition inverse algorithm, my inversion method proposes a new model norm, which incorporates two important weighting functions. One is the L0 quasi norm (enforcing sparse constraints), and the other is depth‐weighting that counteracts the influence of source depth on the resulting potential field data of the solution. Sparseness constraints are imposed by using the L0 quasinorm on model parameters. To solve the representation problem, an L0 quasinorm minimisation model with different smooth approximations is proposed. Hence, the data space (N) method, which is much smaller than model space (M), combined with the gradient‐projected method, and the model space, combined with the modified Newton method for L0 quasinorm sparse constraints, leads to a computationally efficient method by using an N × N system versus an M × M one because N ? M. Tests on synthetic data and real datasets demonstrate the stability and validity of the L0 quasinorm spare norms inversion method. With the aim of obtaining the blocky results, the inversion method with the L0 quasinorm sparse constraints method performs better than the traditional L2 norm (standard Tikhonov regularisation). It can obtain the focus and sparse results easily. Then, the Bouguer anomaly survey data of the salt dome, offshore Louisiana, is considered as a real case study. The real inversion result shows that the inclusion the L0 quasinorm sparse constraints leads to a simpler and better resolved solution, and the density distribution is obtained in this area to reveal its geological structure. These results confirm the validity of the L0 quasinorm sparse constraints method and indicate its application for other potential field data inversions and the exploration of geological structures.     

水平分层土层系统等效阻尼比的简化计算方法

在实际工程场地中,很多土层可视为水平分层,各层土的物理和力学性质存在差异,其中包括土的振动阻尼比。本文讨论水平分层土层系统的等效阻尼比的近似计算方法,基于5个不同的加权函数推导了10种等效阻尼比的计算公式。通过2个算例,分别以等效阻尼比为参数计算水平分层土层的地震反应,并与准确解相比较,分析了不同等效阻尼比近似计算方法的计算精度。数值结果表明,若等效阻尼比计算方法选择不恰当,会导致土层地震反应的计算结果出现较大误差。针对2种不同类型的水平分层土层,建议采用基于三角形分布的加权函数来计算土层系统的等效阻尼比。     

基于矩阵预处理和BPT-SIRT非线性加权反演成像法

为提高弹性波CT层析成像中反演成像的质量,同时考虑到CT走时方程中系数矩阵的病态性和BPT-SIRT算法重建图像质量较低的问题,提出基于主元加权预处理和对BPT-SIRT算法加权的方法。该方法对BPT得到的初始迭代值进行加权预处理,在CT走时方程中的系数矩阵主元上叠加一个权值,再通过非线性的指数加权函数对射线路径矩阵中的元素进行加权。通过数值模拟和预制含缺陷的钢管混凝土试件对该方法进行验证,结果表明本文的方法具有更好的收敛速度和精度,对缺陷的识别效果更好。      

Spatial transport and spectral transfer of solar wind turbulence composed of Alfvén waves and convective structures II: numerical results

This work follows the paper titled “Spatial transport and spectral transfer of solar wind turbulence composed of Alfvén waves and convective structures I: The theoretical model”, and deals with the detailed physics and numerical solution of a two-component solar wind model, consisting of small-scale Alfvén waves and convected structures. In particular, we present numerical results which qualitatively reflect many of the observed features of the radial and spectral evolution of the turbulent energies, the residual energy, the cross-helicity and Alfvén-ratio in high-speed solar wind streams. These features are the following: the formation of a characteristic “inclined eye”, which evolves between the energy spectra displayed over the frequency axis and tends to close in the radial development of the spectra, a steepening of all spectra towards Kolmogorov-like f^-5/3 spectra, the development of the normalized cross-helicity towards a constant not much less than one and the formation of a “trough” form of the Alfvén ratio with a z-shaped left boundary, By weighting special terms in the equations differently, we can also cast light on the physical role of parametric conversion model terms, wave-structure scattering model terms, nonlinear terms, spherical expansion terms and their effects on the radial evolution of turbulent energies in high-speed solar wind streams.