Magnetic susceptibility inversion method with full tensor gradient data using low-temperature SQUIDs

Full tensor magnetic gradient measurements are available nowadays. These are essential for determining magnetization parameters in deep layers. Using full or partial tensor magnetic gradient measurements to determine the subsurface properties, e.g., magnetic susceptibility, is an inverse problem. Inversion using total magnetic intensity data is a traditional way.Because of di culty in obtaining the practical full tensor magnetic gradient data, the corresponding inversion results are not so widely reported. With the development of superconducting quantum interference devices(SQUIDs), we can acquire the full tensor magnetic gradient data through field measurements. In this paper, we study the inverse problem of retrieving magnetic susceptibility with the field data using our designed low-temperature SQUIDs. The solving methodology based on sparse regularization and an alternating directions method of multipliers is established. Numerical and field data experiments are performed to show the feasibility of our algorithm.

Calculation and correction of magnetic object positioning error caused by magnetic field gradient tensor measurement

Magnetic field gradient tensor measurement is an important technique to obtain position information of magnetic objects. When using magnetic field sensors to measure magnetic field gradient as the coefficients of tensor, field differentiation is generally approximated by field difference. As a result, magnetic objects positioning by magnetic field gradient tensor measurement always involves an inherent error caused by sensor sizes, leading to a reduction in detectable distance and detectable angle. In this paper, the inherent positioning error caused by magnetic field gradient tensor measurement is calculated and corrected by iterations based on the systematic position error distribution patterns. The results show that, the detectable distance range and the angle range of an ac magnetic object（2.44 Am^2@1 kHz） can be increased from（0.45 m, 0.75 m）,（0？, 25？） to（0.30 m, 0.80 m）,（0？,80？）, respectively.

Direct numerical simulation study of the interaction between the polymer effect and velocity gradient tensor in decaying homogeneous isotropic turbulence

Direct numerical simulation of decaying homogeneous isotropic turbulence （DHIT） of a polymer solution is performed. In order to understand the polymer effect on turbulence or additive-turbulence interaction, we directly investigate the influence of polymers on velocity gradient tensor including vorticity and strain. By visualizing vortex tubes and sheets, we observe a remarkable inhibition of vortex structures in an intermediate-scale field and a small-scale field but not for a large scale field in DHIT with polymers. The geometric study indicates a strong relevance among the vorticity vector, rate-of-strain tensor, and polymer conformation tensor. Joint probability density functions show that the polymer effect can increase ＂strain generation resistance＂ and ＂vorticity generation resistance＂, i.e., inhibit the generation of vortex sheets and tubes, ultimately leading to turbulence inhibition effects.

The effect of different number of diffusion gradients on SNR of diffusion tensor-derived measurement maps

Diffusion tensor imaging (DTI) is mainly applied to white matter fiber tracking in human brain, but there is still a debate on how many diffusion gradient directions should be used to get the best results. In this paper, the performance of 7 protocols corresponding to 6, 9, 12, 15, 20, 25, and 30 noncollinear number of diffusion gradi-ent directions (NDGD) were discussed by com-paring signal-noise ratio (SNR) of tensor- de-rived measurement maps and fractional ani-sotropy (FA) values. All DTI data (eight healthy volunteers) were downloaded from the website of Johns Hopkins Medical Institute Laboratory of Brain Anatomi-cal MRI with permission. FA, apparent diffusion constant mean (ADC-mean), the largest eigen-value (LEV), and eigenvector orientation (EVO) maps associated with LEV of all subjects were calculated derived from tensor in the 7 proto-cols via DTI Studio. A method to estimate the variance was presented to calculate SNR of these tensor-derived maps. Mean ±standard deviation of the SNR and FA values within re-gion of interest (ROI) selected in the white mat-ter were compared among the 7 protocols. The SNR were improved significantly with NDGD increasing from 6 to 20 (P<0.05). From 20 to 30, SNR were improved significantly for LEV and EVO maps (P<0.05), but no significant dif-ferences for FA and ADC-mean maps (P>0.05). There were no significant variances in FA val-ues within ROI between any two protocols (P>0.05). The SNR could be improved with NDGD in-creasing, but an optimum protocol is needed because of clinical limitations.

ON SHRINKING GRADIENT RICCI SOLITONS WITH POSITIVE RICCI CURVATURE AND SMALL WEYL TENSOR

We show that closed shrinking gradient Ricci solitons with positive Ricci curvature and sufficiently pinched Weyl tensor are Einstein. When Weyl tensor vanishes, this has been proved before but our proof here is much simpler.

基于伪磁梯度张量混合方向tilt梯度的磁源定位方法及其应用:以内蒙古塔木素铀矿床为例

斜磁化磁源的准确定位是磁法勘探有效圈定场源分布及划分断裂构造的重要依据.但受磁化方向影响,总磁异常与地质体分布并无明显对应关系.为此,本文从总磁异常垂向积分出发,推导出了伪磁梯度张量,定义了伪磁梯度张量的混合方向tilt梯度及混合方向解析信号振幅,提出了基于伪磁梯度张量混合方向tilt梯度水平导数组合的场源定位方法.模型试验表明,相对于解析信号振幅、tilt梯度的总水平导数及方向tilt梯度的总水平导数模这3种受磁化方向影响较小的方法,本文提出的磁源定位方法能够更有效地反映出不同磁化方向的磁源位置,具有更强的识别能力和定位精度.在内蒙古塔木素铀矿床的航磁资料应用中,新方法较好地展示出了已知断裂带的展布及出露岩浆岩的分布,同时有效地提取了塔木素铀矿床上方的弱磁异常信息,处理结果为研究区圈定隐伏岩体、划分隐伏断裂及扩大铀矿找矿靶区提供了依据.

基于U⁃Rnet的重力全张量梯度数据反演

重力反演是通过地表信息获取地下地质体空间结构与物理性质的重要手段之一。每个重力梯度分量反映不同的地质体信息,联合重力梯度分量进行重力反演能够更好地研究地下密度异常体的形态和分布。为此,提出基于神经网络的重力全张量梯度数据反演算法,将U⁃Rnet网络应用于重力全张量数据的三维反演问题。为了检验该算法的有效性,采用六种典型模型进行模拟实验,获得了具有清晰边界和稀疏的反演结果。首先,对比L2和Tversky两种损失函数的反演结果,后者的反演结果能更清晰地反映模型的边界位置;然后,对不同梯度张量组合进行反演,四组实验结果在三个方向(x、y、z)上具有不同的反演精度,组合四的误差最低;最后,将该方法应用于美国德克萨斯州文顿盐丘的FTG数据,反演结果与实际地质信息基本吻合。

利用重力梯度张量等位面曲率的地质体定位

针对重力梯度张量曲率的研究,前人的工作主要集中在重力张量曲率的解释及边缘检测中,几乎没有涉及到地下密度异常体的定位.本文结合重力矢量和重力梯度张量提出了一套基于重力梯度张量等位面曲率的地下密度异常体位置估计策略.首先,从重力梯度张量等位面曲率的基本定义出发,计算重力梯度张量等位面曲率.然后,通过寻找球面或圆形等位面的重力梯度张量曲率,提出了利用最大主曲率定位地下密度异常体位置的源参数估计方法,并详细推导了估计3D球体(质点)和2D水平线源位置信息的解析表达式.再者,针对噪声和多源存在的情况,提出了一套利用重力梯度张量等位面曲率获得密度异常体位置信息的稳健估计流程,并利用模糊C均值聚类算法进一步确定地下密度异常体的中心位置.最后,通过理论模型测试和文顿盐丘实测航空重力梯度数据测试,验证了本文算法的可行性和可靠性.结果表明:在满足曲率半径定义条件的情况下,本文所提出的源参数估计方法可以定位单个或多个地下3D和2D密度异常体的空间位置,具有较好的稳健性和抗噪能力.该方法拓展了重力梯度张量曲率的应用范围,可为重力梯度张量的三维反演工作提供先验的空间位置信息.

修正共轭梯度算法求解四元数Sylvester张量方程

本文提出一类张量形式的修正共轭梯度算法求解四元数Sylvester张量方程.证明在不计舍入误差的情况下,所提方法可在有限迭代步内获得张量方程组的解.进一步,通过选择特殊类型的初始张量,可获得方程组的唯一极小Frobenius范数解.通过数值算例验证了所提出算法的可行性和有效性.

拖曳式海洋三分量磁力梯度仪的数据处理方法与应用

拖曳式海洋三分量磁力梯度仪能够同时获得地磁三分量和磁力梯度数据,与传统的拖曳式总场磁力仪相比具有降低船磁干扰和抵抗地磁日变影响等优点,也存在灵敏度误差、零点偏移误差、正交性误差和位置误差等校准处理的难点。在实际航次中用拖曳式海洋三分量磁力梯度仪实测得到一条剖面数据,经校准后与G880磁力仪重复线测量的地磁总场数据进行外符合精度评价,表明其具有较高的稳定性和可靠性。根据三分量磁力梯度数据计算得到其张量不变量以及磁边界走向图,并结合欧拉反褶积计算,对测线上的磁源体进行了有效识别和解释。结果显示拖曳式海洋三分量磁力梯度仪能够有效获得地磁总场、分量及梯度等多参量信息,可为海洋地磁场测量提供更有效的技术手段。

Forward Modeling of Gravity,Gravity Gradients,and Magnetic Anomalies due to Complex Bodies

On the basis of the results of improved analytical expression of computation of gravity anomalies due to a homogeneous polyhedral body composed of polygonal facets, and applying the forward theory with the coordinate transformation of vectors and tensors, we deduced both the analytical expressions for gravity gradient tensors and for magnetic anomalies of a polygon, and obtained new analytical expressions for computing vertical gradients of gravity anomalies and vertical component of magnetic anomalies caused by a polyhedral body. And also we developed explicitly the complete unified expressions for the calculation of gravity anomalies, gravity gradient, and magnetic anomalies due to the homogeneous polyhedron. Furthermore, we deduced new analytical expressions for computing vertical gradients of gravity anomalies due to a finite rectangular prism by applying the newly obtained expressions for gravity gradient tensors due to a polyhedral target body. Comparison with forward calculation of models shows the correctness of these new expressions. It will reduce forward calculation time of gravity-magnetic anomalies and improve computational efficiency by applying our unified expressions for joint forward modeling of gravity-magnetic anomalies due to homogeneous polyhedral bodies.

Joint inversion of gravity and multiple components of tensor gravity data

Geological structures often exhibit smooth characteristics away from sharp discontinuities. One aim of geophysical inversion is to recover information about the smooth structures as well as about the sharp discontinuities. Because no specific operator can provide a perfect sparse representation of complicated geological models, hyper-parameter regularization inversion based on the iterative split Bregman method was used to recover the features of both smooth and sharp geological structures. A novel preconditioned matrix was proposed, which counteracted the natural decay of the sensitivity matrix and its inverse matrix was calculated easily. Application of the algorithm to synthetic data produces density models that are good representations of the designed models. The results show that the algorithm proposed is feasible and effective.

Incompatible deformation field and Riemann curvature tensor

Compatibility conditions of a deformation field in continuum mechanics have been revisited via two different routes. One is to use the deformation gradient, and the other is a pure geometric one. Variations of the displacement vector and the displacement density tensor are obtained explicitly in terms of the Riemannian curvature tensor. The explicit relations reconfirm that the compatibility condition is equivalent to the vanishing of the Riemann curvature tensor and reveals the non-Euclidean nature of the space in which the dislocated continuum is imbedded. Comparisons with the theory of Kr¨oner and Le-Stumpf are provided.

基于归一化特征值的重磁数据边界识别方法及其在西藏古堆地热勘探中的应用

基于重磁数据的地质体边界识别是地球物理数据解释的重要内容。随着重磁梯度张量测量技术的不断发展,基于梯度张量数据圈定地质体构造边界位置的方法也在不断发展。本文定义一种基于梯度张量特征值的归一化边界识别滤波器,可同时进行不同深度目标体的边界位置圈定。为验证本文方法的有效性,将建立方法应用于合成的重磁数据中,相比于传统边界识别滤波器,本文滤波器具有更高的水平分辨率和精度,并且在正负叠加异常的试验中,可有效地避免虚假边界的产生。最后,将归一化特征值边界识别滤波器应用于西藏古堆地热田的地质构造解释,结果表明,基于特征值的边界圈定方法能够更准确地显示地质构造的边界位置,发现更多地质体构造细节特征。

基于改进梯度下降算法优化的磁梯度张量组合不变量算法的管道缺陷边缘识别模型

针对油气管道磁记忆检测受方向影响较大且缺陷边缘精确识别困难的问题,提出了一种基于改进梯度下降算法(MGD)优化的磁梯度张量组合不变量算法模型,用于管道缺陷边缘的精确识别。以L245N管线钢为试验材料,预制不同深度、不同直径的圆孔状缺陷,设计磁梯度张量测量系统,结合TSC-5M-32型磁记忆仪进行检测实验,获得管道的磁梯度张量矩阵。为克服检测方向对磁记忆信号的影响,分别提取磁梯度张量第二、第三不变量I_(1)、I_(2),进一步考虑这两种不变量在缺陷边缘处易出现模糊,根据Cardano公式对两种不变量进行改进,并分别设置权值a、b进行叠加获得组合不变量I,利用分数阶求导改进梯度下降算法确定最优权值,建立管道缺陷边缘磁记忆识别模型。研究结果表明:该模型对缺陷边缘识别平均相对误差为3.59%,最大相对误差为6%,为实际工程中管道缺陷边缘精准识别提供了可行办法。

GRADIENT ESTIMATES FOR POSITIVE SMOOTH f-HARMONIC FUNCTIONS

For Riemannian manifolds with a measure, we study the gradient estimates for positive smooth f-harmonic functions when the ∞-Bakry-Emery Ricci tensor and Ricci tensor are bounded from below, generalizing the classical ones of Yau （i.e., when ： is constant）.

空间—波数域二维强磁场及其梯度张量数值模拟研究

基于磁位泊松方程提出了一种空间—波数混合域二维数值模拟方法。该方法将二维强磁性体磁位满足的偏微分方程通过一维傅里叶变换转化为不同波数之间相互独立的常微分方程,即将二维数值模拟问题转换为一维数值模拟问题,并利用一维有限元法进行求解,可大大降低存储需求和计算量。对求得的强磁场利用紧算子进行迭代以获得高精度近似解。算法中保留垂向为空间域,可根据需求对网格进行灵活剖分,有利于复杂条件下的模拟。另外,不同波数之间常微分方程相互独立,具有较好的并行性。算法充分利用了傅里叶变换的快速性和迭代算法的稳定性,实现了强磁场及其梯度张量的二维数值模拟。设计三个模型(水平地形下的圆柱体模型、棱柱体模型及起伏地形模型)进行计算,对文中方法数值解与解析解、COMSOL软件所得的数值解进行对比,结果表明该算法计算速度快、精度高,适用于起伏地形条件下的数值计算。

基于Tensor Train分解的Sylvester张量方程求解

为克服Sylvester张量方程高阶情形带来的维数灾难,应用张量的Tensor Train(TT)分解来减少参数的存储和计算复杂度,并结合张量TT分解的特点,提出一种交替随机梯度法用于求解Sylvester张量方程,给出收敛性分析,最后通过数值实验验证了所提算法的有效性。

空间自适应方向全变分地震数据去噪模型

随着我国油气勘探工作的不断推进,地震勘探面临着重大挑战。受到复杂的勘探环境、采集方式、检波器灵敏度等因素的影响,野外采集的地震数据中往往混杂着大量随机噪声,导致后续地震数据处理的保真度、信噪比和分辨率降低,并且最终影响地质解释的精确性、可靠性。为了突破传统地震数据处理问题的局限性,提出了一种用于地震数据随机噪声压制的空间自适应方向全变分正则化模型。首先,针对地震反射同相轴具有空间变化的方向性和倾角计算抗噪性差的问题,提出了基于梯度结构张量的空变倾角逐点估计公式来获取同相轴的方向信息;然后,建立空间自适应方向全变分地震数据去噪模型,并采用优化最小化算法求解模型;最后,讨论了该模型的参数选取方法,将合成地震数据和实际地震数据的去噪结果与同类方法进行比较。实验结果表明,所提出的模型不但能较好地提高地震剖面的垂直分辨率和同相轴的横向连续性,而且在提高信噪比的同时能够保留更多的地质特征信息。

Hamilton-Souplet-Zhang's gradient estimates for two weighted nonlinear parabolic equations

In this paper, we consider gradient estimates for positive solutions to the following weighted nonlinear parabolic equations on a complete smooth metric measure space with only Bakry-Émery Ricci tensor bounded below: One is $${u_t} = {\Delta _f}u + au\log u + bu$$ with a, b two real constants, and another is $${u_t} = {\Delta _f}u + \lambda {u^\alpha }$$ with λ, α two real constants. We obtain local Hamilton-Souplet-Zhang type gradient estimates for the above two nonlinear parabolic equations. In particular, our estimates do not depend on any assumption on f.