用变分玻恩迭代方法重建二维非均匀介质结构

提出了用于二维轴对称非均匀介质结构的反演和成像的一种新的反演迭代方法──变分玻恩迭代方法（ＶＢＩＭ）．首先利用玻恩近似将非线性积分方程线性化,然后应用变分方法导出用于反演的电场积分方程．正演数据则利用高效的数值模式匹配方法获得．数值结果表明,ＶＢＩＭ与ＢＩＭ相比,其收敛速度、成像质量等均得到较大的改善。     

电极型电阻率测井反演算法研究

低频电极型测井仪的响应总是受井眼、围岩和侵入的影响，而传统的测井综合解释无法完全有效地校正这些影响．本文将收敛速度快而效果好的变形波恩迭代（DBIM）方法用于处理轴对称二维非均匀介质分布的电导率反演问题．在每次迭代过程中采用了快速的半解析的模式匹配的正演方法，半解析地表达出反演中所需计算的格林函数的偏导数；并在此基础上半解析地求出了反演的非线性积分方程中的积分运算，大大提高了反演速度和精度．     

感应测井视电导率和真电导率关系的积分方程

本文研究感应测井仪器测量的感应电动势和视电导率的关系、视电导率和地层真电导率的关系.假定格林函数中的电导率是场点坐标的函数,运用格林公式推导出电场强度的视电导率表达式,建立了视电导率和真电导率关系的积分方程,通过对所得积分方程的深入分析,得出等效电导率与视电导率相等以及视电导率函数值包含真电导率两个结论,并提出由井轴视电导率函数求取真电导率的方法,通过数值计算验证了本文提出的方法的有效性.按照上述做法,仪器每移动一点,发射线圈产生一个电场分布,若干接收线圈得到一个视电导率分布,这就形成了感应电场测井.     

用变分坡恩迭代方法重建二维非均匀介质结构

提出了用于二维轴对称非均匀介质结构的反演和成像的一种新的反演迭代方法－变分坡恩迭代方法（ＶＢＩＭ）,首先利用玻恩近似将非线民生积分方程线性化,然后应用变分方法导出用于反演的电场积分方程,正演数据则利用高效的数值模式匹配方法获得,数值结果表明,ＶＢＩＭ与ＢＩＭ相比,其收敛速度,成像质量等均得到较大的改善。     

双侧向电阻率测井反演算法研究

双侧向测井仪因其激励源的复杂性给其测量数据的反演带来了极大的困难．本文利用变形玻恩（Born）迭代法对轴对称二维非均匀介质中双侧向测井仪的位场数据进行了严格反演．在每一次迭代过程中采用了高效数值模式匹配的正演计算方法,它可半解析地求出反演中所需计算的格林（Green）函数及其偏导数,同时利用所推导出的位场非线性积分方程中积分运算的半解析形式,大大提高了计算效率和反演质量．通过反演实例表明,反演结果与电导率原状剖面吻合较好,并且受噪声影响较小．     

井间电磁成像的迭代反演算法

提出一种基于逐次逼近解法的选代反演算法,对并间地层电导率的构造进行成像．该方法用一阶Ｂｏｒｎ近似将积分方程线性化,得到对电导率分布的初始估计,在选代反演中用高阶Ｂｏｒｎ近似对井间地层电导率构造进行更精确的估计．应用该方法还可以对井间电子率分布进行二次成像,从而使成像分辨率更为准确．数值计算结果表明,这种迭代反演算法与基于Ｂｏｒｎ近似、二阶Ｂｏｒｎ近似和扩展Ｂｏｒｎ近似的反演算法相比提高了成像分辨率,且计算效率相当．     

地层各向异性指数因子的计算方法

针对三维感应测井问题,提出了利用三维感应测井分量视电导率确定地层(TI地层)各向异性指数因子(水平电导率与垂直电导率之比)的计算与确定方法.论文分别给出了图版法和牛顿迭代计算方法,介绍了地层各向异性指数因子的图板制作及利用图版确定地层各向异性指数因子的方法;导出了地层视电导率偏导计算方程,利用非线性方程组的牛顿迭代解法实现了地层各向异性指数因子的数值计算,还给出了数值算例,计算结果与地层模型获得了很好的一致性,表明本方法对分析地层的各向电导特性及测井资料解释应用将具有重要的借鉴和参考意义.     

三维井间电磁场的正反演计算

提出了一种计算体积分方程的方法－改进型局域非线性迭代（MLNI），并用其对三维井间电磁场进行反演计算，该方法将井间大尺度散射体分为近场和远场区域两部分，它们的位置和尺寸均随场点位置的变化而改变，采用局域非线性近似计算近场区域的影响，将远场区域的影响作为外部激励源，采用迭代方法计算，该方法具有计算速度快，所需内存量少，收敛性好的优点，在反演中采用基于MLNI的非线性反演方法，利用这种方法可以将成像区域集中于一定范围内而不是整个三维空间，该反演方法由于考虑到了雅可比矩阵元素的非线性项，因而与传统的Born迭代反演方法相比更为精确和稳定，考虑到信息量和计算机内存的限制，第1次成像采用双重体元分割法进行较粗略的成像，然后缩小成像范围进行第2次较精确的成像，数值计算结果表明，MLNI是一种有效的计算井间大尺度异常体散射场的方法，将该方法用于反演过程能够得到较高分辨率的三维井间电导率图像。     

二维电阻率成像研究

电阻率成像中最关键的问题就是获得雅可比偏导数矩阵。本文从二维微分方程的积分解出发推导了一种新的电阻率成像的雅可比偏导数矩阵,同时形成了成像方程。用内外迭代相结合的高斯塞德儿迭代方法解成像方程可以得到电阻率的分布图像。数值模拟结果表明该方法是有效和可靠的,尤其值得注意的是积分法电阻率成像方法初始模型可以采用均匀模型,减小了对初始模型的依赖。对用其它方法难以获得好的成像结果的单一高阻体,积分法也得到了较好的成像结果。河南商丘某野外资料结果表明,成像结果和实际地质情况吻合较好。     

边界积分方程用于电阻率Zohdy反演的初步研究

研究利用边界积分方程进行电阻率Ｚｏｈｄｙ反演的有关技术,结果表明在探测区域划分为数千个单元的情况下,为节省正、反演过程中所需要的计算机内存和 ＣＰＵ时间,可采取以下措施：（１）假定电位分块线性变化,可以使节点数目显著减少,这里每一块都包含若干个单元;（２）利用边界积分方程的特点,可事先计算并存储所有必要的边界积分值,供每一步迭代中调用;（３）利用边界积分方程所特有的数值延拓功能,可以进一步减少节点数目．为了说明这些手段的有效性,先给出了一个数值模拟成像结果,它清楚地表明基于边界积分方程的电阻率 Ｚｏｈｄｙ反演是快速而有效的,然后给出一个实测剖面的成像实例,成像结果经钻孔检验正确．     

AN EXTENSION OF THE BORN INVERSION METHOD TO A DEPTH DEPENDENT REFERENCE PROFILE*

An extension of the multidimensional Born inversion technique for acoustic waves is described. In earlier work, a perturbation in reference sound velocity was determined by assuming that the reference velocity was constant. In this extension, we allow the reference velocity to be a function of the depth variable z. The output of this method is a high-frequency bandlimited reflectivity function of the subsurface. The reflectivity function is an array of bandlimited singular functions scaled by the normal reflection strength. Each singular function is a Dirac delta function of a scalar argument which measures distance normal to a reflecting interface. Thus, the reflectivity function is an indicator map of subsurface reflectors equivalent to the map produced by migration. In addition to the assumption of small perturbation, the method requires that the reflection data reside in the high frequency regime in a well-defined sense. The method is based on the derivation of an integral equation for the perturbation in sound velocity from a known reference velocity. When the reference velocity is constant, the integral equation admits an analytic solution as a multifold integral of the reflection data. Further high frequency asymptotic analysis simplifies this integral considerably and leads to an extremely efficient numerical algorithm for computing the reflectivity function. The development of a computer code to implement this constant-reference-velocity solution is published elsewhere. For a reference velocity c(z) we can no longer invert the integral equation exactly. However, we can write down an asymptotic high-frequency approximation for the kernal of the integral equation and an asymptotic solution for the perturbation. The computer implementation of this result is designed along the same lines as the code for constant background velocity. In tests the total processing time for this algorithm with depth-dependent background velocity is usually considerably less than that required by a standard Kirchhoff migration algorithm. The method is implemented as a migration technique and compared with alternative migration algorithms on the flanks of the salt dome.     

CALCULATION OF GALVANIC EFFECTS BY MEANS OF THE METHOD OF SUB-AREAS*

A numerical method is proposed for solving the problem of steady current flow. The electrodynamic model is replaced by the equivalent stationary charge distribution obtained by Poisson's analysis, in which the surface integral equation for field intensity is reduced to a set of simultaneous linear algebraic equations by means of the method of sub-areas. The solution of the set allows the calculation of an approximation for the charge density distribution on the discontinuity surfaces of conductivity. The method is valid for complex conductivities, whereby the apparent phase shift of IP can be calculated from the complex potential or field intensity. The phase shift anomaly calculated as an application is very similar to the corresponding frequency effect anomaly. The method allows the calculation of the mise-à-la-masse effect as a solution to a potential problem, in which the primary current electrode is located within the body to be surveyed.     

POTENTIAL FIELD OF A STATIONARY ELECTRIC CURRENT USING FREDHOLM'S INTEGRAL EQUATIONS OF THE SECOND KIND*

An integral equation method is described for solving the potential problem of a stationary electric current in a medium that is linear, isotropic and piecewise homogeneous in terms of electrical conductivity. The integral equations are Fredholm's equations of the ‘second kind’ developed for the potential of the electric field. In this method the discontinuity-surfaces of electrical conductivity are divided into ‘sub-areas’ that are so small that the value of their potential can be regarded as constant. The equations are applied to 3-D galvanic modeling. In the numerical examples the convergence is examined. The results are also compared with solutions derived with other integral equations. Examples are given of anomalies of apparent resistivity and mise-a-la-masse methods, assuming finite conductivity contrast. We show that the numerical solutions converge more rapidly than compared to solutions published earlier for the electric field. This results from the fact that the potential (as a function of the location coordinate) behaves more regularly than the electric field. The equations are applicable to all cases where conductivity contrast is finite.     

Seismic inversion with generalized Radon transform based on local second-order approximation of scattered field in acoustic media

Sound velocity inversion problem based on scattering theory is formulated in terms of a nonlinear integral equation associated with scattered field. Because of its nonlinearity, in practice, linearization algorisms (Born/single scattering approximation) are widely used to obtain an approximate inversion solution. However, the linearized strategy is not congruent with seismic wave propagation mechanics in strong perturbation (heterogeneous) medium. In order to partially dispense with the weak perturbation assumption of the Born approximation, we present a new approach from the following two steps: firstly, to handle the forward scattering by taking into account the second-order Born approximation, which is related to generalized Radon transform (GRT) about quadratic scattering potential; then to derive a nonlinear quadratic inversion formula by resorting to inverse GRT. In our formulation, there is a significant quadratic term regarding scattering potential, and it can provide an amplitude correction for inversion results beyond standard linear inversion. The numerical experiments demonstrate that the linear single scattering inversion is only good in amplitude for relative velocity perturbation ( \( \delta_{c}/c_{0} \) ) of background media up to 10 %, and its inversion errors are unacceptable for the perturbation beyond 10 %. In contrast, the quadratic inversion can give more accurate amplitude-preserved recovery for the perturbation up to 40 %. Our inversion scheme is able to manage double scattering effects by estimating a transmission factor from an integral over a small area, and therefore, only a small portion of computational time is added to the original linear migration/inversion process.     

基于流变体模型的波恩近似方程地震正演模拟

常规τ值法假设应力松弛时间与应变延迟时间近似相等, 造成了常Q模型拟合精度低. 本文利用精确的广义流变体模型Q值计算公式, 研究改进的τ值法求解常Q模型参数. 根据地震波散射理论, 推导了基于广义流变体模型的黏滞性介质一阶波恩近似方程, 结合位移-速度关系得到了含卷积完全匹配层边界条件的黏滞性介质应力-速度方程的一阶波恩近似表达式. 通过数值实验验证并对比了黏滞性介质中全波波动方程、 一阶波恩近似方程以及单程波波动方程的波场特征, 讨论了基于流变体模型的黏滞性介质一阶波恩近似方程对速度扰动和Q扰动的适应性, 以及对旅行时和振幅精度的影响.      

Constraining of the zero total surface charge in galvanic modelling

The galvanic problem is frequently solved by a Fredholm integral equation of the second kind based on a single layer source formulation. At higher conductivity contrasts between the model and its surroundings the homogeneous part of the integral equation approaches an eigenvalue equation. With infinite contrast the solution of this limiting integral equation is non-unique, but in the subspace of zero total charge the solution is unique. This mathematical property of the integral equation is reflected in its numerical solution with the result that large numerical errors may appear and convergence of the solution becomes very slow. Errors are, for the most part, related to the computed excess charge generated in the numerical solution. The effect is studied by comparing the results computed from the solution of the integral equation alone with those computed from a particular solution where the requirement of zero total charge is used as a constraint. The model examples clearly show that the use of the constraint condition significantly improves the accuracy of the results.     

一种新的三维大地电磁积分方程正演方法

采用规则六面体单元和并矢Green函数奇异积分等效积分技术,已有的大地电磁积分正演方法具有不能有效模拟地下复杂地质体和计算精度偏低的缺点.本文提出了一种新的三维大地电磁积分方程正演技术,即采用四面体单元、解析的并矢Green函数奇异积分表达式,达到既能模拟地下复杂异常体,又能有效提高已有积分方程法计算精度的目的.首先,采用四面体网格技术离散地下复杂异常体,获得四面体单元上的大地电磁积分方程.然后,利用针对四面体单元开发的新的奇异值积分的解析表达式,准确计算线性方程中的并矢Green函数的奇异积分,从而获得精确的线性方程.借助于PARDISO高性能并行直接求解器,实现了三维大地电磁问题的高精度求解.最后,基于国际标准3D-1模型和六棱柱模型,通过与其他方法结果的对比分析,验证了本文方法的正确性、处理高电导率对比度的能力(1000:1)和处理复杂模型的能力.