Unstructured grid based 2-D inversion of VLF data for models including topography

We present a 2-D inversion code incorporating a damped least-squares and a minimum-model approach for plane wave electromagnetic (EM) methods using an adaptive unstructured grid finite element forward operator. Unstructured triangular grids permit efficient discretization of arbitrary 2-D model geometries and, hence, allow for modeling arbitrary topography. The inversion model is parameterized on a coarse parameter grid which constitutes a subset of the forward modeling grid. The mapping from parameter to forward modeling grid is obtained by adaptive mesh refinement. Sensitivities are determined by solving a modified sensitivity equation system arising from the derivative of the finite element equations with respect to the model parameters. Firstly, we demonstrate that surface topography may induce significant effects on the EM response and in the inversion result, and that it cannot be ignored when the scale length of topographic variations is in the order of magnitude of the skin depth. Secondly, the dependency of the inversion on the starting model is discussed for VLF and VLF-R data. Thirdly, we demonstrate the inversion of a synthetic data set obtained from a model with topography. Finally, the inversion approach is applied to field data collected in a region with undulating topography.     

Two-dimensional joint inversion of radiomagnetotelluric and direct current resistivity data

An algorithm for the two-dimensional (2D) joint inversion of radiomagnetotelluric and direct current resistivity data was developed. This algorithm can be used for the 2D inversion of apparent resistivity data sets collected by multi-electrode direct current resistivity systems for various classical electrode arrays (Wenner, Schlumberger, dipole-diplole, pole-dipole) and radiomagnetotelluric measurements jointly. We use a finite difference technique to solve the Helmoltz and Poisson equations for radiomagnetotelluric and direct current resistivity methods respectively. A regularized inversion with a smoothness constrained stabilizer was employed to invert both data sets. The radiomagnetotelluric method is not particularly sensitive when attempting to resolve near-surface resistivity blocks because it uses a limited range of frequencies. On the other hand, the direct current resistivity method can resolve these near-surface blocks with relatively greater accuracy. Initially, individual and joint inversions of synthetic radiomagnetotelluric and direct current resistivity data were compared and we demonstrated that the joint inversion result based on this synthetic data simulates the real model more accurately than the inversion results of each individual method. The developed 2D joint inversion algorithm was also applied on a field data set observed across an active fault located close to the city of Kerpen in Germany. The location and depth of this fault were successfully determined by the 2D joint inversion of the radiomagnetotelluric and direct current resistivity data. This inversion result from the field data further validated the synthetic data inversion results.     

Two-dimensional data-space joint inversion of magnetotelluric,gravity, magnetic and seismic data with cross-gradient constraints

In recent years, joint inversion has been widely used for integrated geological interpretation. We extended a data-space joint inversion algorithm of magnetotelluric, gravity and magnetic data to include first-arrival seismic travel-time and normalized cross-gradient constraints. We describe the main features of the algorithm and apply it to synthetic data generated for hypothetical models. For the synthetic data, we find that the joint inversion with multiple parameters is superior to the joint inversion with two or even three parameters, which can reduce the multisolution of inversion results more effectively. Furthermore, data-space joint inversion involves fewer memory requirements and better calculation speeds than traditional model-space joint inversion. The normalized cross-gradient constraints can better couple model parameters of different magnitudes compared with traditional unnormalized cross-gradient constraints, resulting in higher levels of structural similarity among resistivity, density, magnetic susceptibility and velocity models.     

Two-dimensional inversion of spectral induced polarization data using MPI parallel algorithm in data space

Traditional two-dimensional (2D) complex resistivity forward modeling is based on Poisson’s equation but spectral induced polarization (SIP) data are the coproducts of the induced polarization (IP) and the electromagnetic induction (EMI) effects. This is especially true under high frequencies, where the EMI effect can exceed the IP effect. 2D inversion that only considers the IP effect reduces the reliability of the inversion data. In this paper, we derive differential equations using Maxwell’s equations. With the introduction of the Cole–Cole model, we use the finite-element method to conduct 2D SIP forward modeling that considers the EMI and IP effects simultaneously. The data-space Occam method, in which different constraints to the model smoothness and parametric boundaries are introduced, is then used to simultaneously obtain the four parameters of the Cole—Cole model using multi-array electric field data. This approach not only improves the stability of the inversion but also significantly reduces the solution ambiguity. To improve the computational efficiency, message passing interface programming was used to accelerate the 2D SIP forward modeling and inversion. Synthetic datasets were tested using both serial and parallel algorithms, and the tests suggest that the proposed parallel algorithm is robust and efficient.     

Resolution of airborne VLF data

The interpretation of airborne VLF data represents an important aspect of geophysical mapping of the upper few hundred meters of the Earth's crust, especially in areas with crystalline rocks. We have examined the ability of the single frequency VLF method to provide quantitative subsurface resistivity information using two generic models and standard airborne parameters with a flight altitude of 70 m and a frequency of 16 kHz. The models are long thin conductor (10 m thick, 10 Ω m resistivity and 1 km long) and a wider buried conductive dike (100 Ω m resistivity and 500 m wide). Using standard regularized inversion it turned out that for both models the conductivity of the conductors are underestimated and the vertical resolution is rather poor. The lateral positions of the minimum of the resistivity distributions coincide well with the true positions of the shallow conductors. For deeper conductors the position of the minimum resistivity moves from the edges of the conductor into the conductor. The depth to the minimum of the resistivity anomalies correlates well with the true depth to the top of the conductors although the latter is always smaller than the former.Interpretation of field airborne data collected at 70 m flight height resolved both small scale and large scale near surface conductors (conductance ∼1 S). Deeper conductors show up in the VLF data as very long wavelength anomalies that are particularly powerful in delineating the lateral boundaries of the conductors. Many of the VLF anomalies in the Stockholm area are dominated by these deep conductor responses with some near surface conductors superimposed. The deep conductors often follow topographic lows coinciding with metasediments. We interpret the frequent absence of near surface responses at 70 m flight height as a result of weak coupling between the primary VLF wave and the small scale (in all three dimensions) near-surface conductors.Radio magnetotelluric (RMT) ground measurements were carried out along a short profile coinciding with part of an airborne profile. Using data at 9 frequencies (14–250 kHz) small scale conductors in the upper few tens of meters, not identified from the airborne data, could be well resolved. Large scale deeper conductors could be identified by both methods at nearly the same positions.     

基于交叉梯度约束的时间域激发极化法二维同步反演

时间域激发极化法(Time-domain induced polarization method,简称为TDIP)已有的反演算法采用的是分步反演的思路,即先由视电阻率资料反演电阻率,固定电阻率再由视极化率资料反演极化率,这样就存在极化率结果严重依赖于电阻率反演结果的问题.为了有效解决这一问题,本文实现了TDIP二维数据空间分步反演算法,提出了基于交叉梯度约束的TDIP二维同步反演策略,实现了交叉梯度约束的电阻率和极化率二维同步反演算法.分别用电阻率和极化率结构一致和不一致的二维模型合成数据进行了分步和同步反演试算,对不同模型试算结果进行了对比分析.结果表明:对于电阻率和极化率结构一致和不一致模型,同步反演结果比分步反演结果能更好地确定异常体的空间分布范围,反演得到的电阻率和极化率值更接近真值.理论模型算例表明本文提出的同步反演算法有效解决了分步反演的问题,优于分步反演算法,具有更好的实用性.     

大地电磁资料精细处理和二维反演解释技术研究(六)——交错模型的大地电磁二维反演

交错模型指的是由几组走向不同的线性构造（近似二维）在空间（纵向、横向）上组合而成的模型.对于电性结构,交错模型表现为纵向或横向上的构造存在电性主轴方位的变化.从构造维性的角度来看,交错模型是一种由二维模型组合而成的特殊的三维模型,直接对其进行大地电磁二维反演,不易获得可靠的反演结果.本文针对交错模型的特点,提出分频段-分区段反演方案.该方案首先需要借助于阻抗张量成像技术,在频率域确定组成交错模型的各线性构造的电性主轴,然后,针对不同的频段、区段,选择对应电性主轴的数据进行反演,通过初始模型的构建将不同电性主轴方位的反演结果对接起来.本文通过三维理论模型的研究,系统展示了分频段-分区段反演的全过程,归纳得到：在分频段反演时先做低频段反演,在分区段反演时先做二维性更不显著一侧的反演.最后,本文将这一技术用于郯庐断裂带中南段一条实测剖面的反演中,其结果与常规二维反演结果相比较,深部的信息更为丰富,且与其他已有地质、地球物理结果的可对比性更好,表明在构造复杂地区,大地电磁分频段-分区段二维反演具有较高的模型分辨率和可靠性.     

基于贝叶斯方法的二维大地电磁尖锐边界反演研究

针对常规大地电磁（Magnetotelluric,MT）反演方法对电阻率异常体边界不太敏感的问题,本文尝试基于贝叶斯理论开展二维大地电磁电阻率尖锐边界反演研究.在反演中,模型参数由边界位置及内部电阻率组成,通过贝叶斯理论将模型参数与数据相联系,采用Markov Chain Monte Carlo（MCMC）的Metropolis-Hastings（MH）方法对后验概率密度函数（Posteriori Probability Density,PDD）进行采样.采样过程中无罚值函数约束,完全以数据自身所包含的信息对模型进行约束,同时与有限约束进行比较,并考虑不同起始采样点对结果的影响.以接受率为参考,用模型算例说明MH方法中建议分布函数选择的重要性.当模型参数间相关性较弱时,使用边缘概率分布对采样结果进行分析.该方法能给出模型参数的分布范围,并给出该模型参数范围对应的数据范围.通过与已知模型的对比及数据拟合情况分析检验了该反演方法的有效性.该方法有助于提高大地电磁尖锐边界反演的分辨能力.

     

Two-dimensional inversion of magnetotelluric data with consecutive use of conjugate gradient and least-squares solution with singular value decomposition algorithms

I investigated the two‐dimensional magnetotelluric data inversion algorithms in studying two significant aspects within a linearized inversion approach. The first one is the method of minimization and second one is the type of stabilizing functional used in parametric functionals. The results of two well‐known inversion algorithms, namely conjugate gradient and the least‐squares solution with singular value decomposition, were compared in terms of accuracy and CPU time. In addition, magnetotelluric data inversion with various stabilizers, such as L2‐norm, smoothing, minimum support, minimum gradient support and first‐order minimum entropy, were examined. A new inversion algorithm named least‐squares solution with singular value decomposition and conjugate gradient is suggested in seeing the outcomes of the comparisons carried out on least‐squares solutions with singular value decomposition and conjugate gradient algorithms subject to a variety of stabilizers. Inversion results of synthetic data showed that the newly suggested algorithm yields better results than those of the individual implementations of conjugate gradient and least‐squares solution with singular value decomposition algorithms. The suggested algorithm and the above‐mentioned algorithms inversion results for the field data collected along a line crossing the North Anatolian Fault zone were also compared each other and results are discussed.     

Petrophysical inversion of AVA data

We investigate the interactions between the elastic parameters, V_P, V_S and density, estimated by non-linear inversion of AVA data, and the petrophysical parameters, depth (pressure), porosity, clay content and fluid saturation, of an actual gas-bearing reservoir. In particular, we study how the ambiguous solutions derived from the non-uniqueness of the seismic inversion affect the estimates of relevant rock properties. It results that the physically admissible values of the rock properties greatly reduce the range of possible seismic solutions and this range contains the actual values given by the well. By means of a statistical inversion, we analyse how approximate a priori knowledge of the petrophysical properties and of their relationships with the seismic parameters can be of help in reducing the ambiguity of the inversion solutions and eventually in estimating the petrophysical properties of the specific target reservoir. This statistical inversion allows the determination of the most likely values of the sought rock properties along with their uncertainty ranges. The results show that the porosity is the best-resolved rock property, with its most likely value closely approaching the actual value found by the well, even when we insert somewhat erroneous a priori information. The hydrocarbon saturation is the second best-resolved parameter, but its most likely value does not match the well data. The depth of the target interface is the least-resolved parameter and its most likely value is strongly dependent on a priori information. Although no general conclusions can be drawn from the results of this exercise, we envisage that the proposed AVA–petrophysical inversion and its possible extensions may be of use in reservoir characterization.     

Magnetotelluric data inversion with seismic data constraint

Introduction The research on the structure and physical property of ancient hidden hill, igneous rocks and basement is relatively difficult by using seismic data only. If we combine seismic data, magneto-telluric (MT) data and geophysical data together, better results can be obtained for the above problem. A number of geophysicists at home and abroad, such as CHEN and WANG (1990), Siri-punvarapor and Egbert (2000) have tried many methods to solve the problem by the inversion of seismic da…     

The inversion of surface wave data

Observations of surface waves depend both on the structure traversed and the nature of the source, and therefore inversion of surface wave data can yield information about both structure and sources. The available methods for structural inversion are compared and discussed and a suggestion made for improved source inversion.     

大地电磁测深反演方法现状与评述

如何通过改进大地电磁测深(MT)反演方法来提高数据解释的精确度一直都是大地电磁测深研究领域的重要课题.本文以作者所在研究组从事的科研项目为依托,介绍了国内外主要的大地电磁反演方法.通过总结将反演方法分为定性近似反演方法、基于目标函数的线性或非线性选代反演方法和全局搜索最优反演方法三大类,并从目标函数构建、灵敏度矩阵计算...     

基于交替方向乘子算法的二维磁异常稀疏反演

磁异常反演是获取地下场源磁化率分布的重要手段之一,在地球勘探中扮演着重要角色.在磁异常反演中,对比光滑反演,稀疏反演的结果具有边界分明,物性参数分布集中的特点,更符合实际情况.针对稀疏反演,本文首先构建了具有代表性的基于L₁范数目标函数,利用交替方向乘子算法可分离凸函数的特点,将极小化L₁范数的优化问题分解为一系列的子问题,通过对子问题求解获得原问题的解;为了增强交替方向乘子算法的适应性,本文结合广义软阈值函数将交替方向乘子法推广于L_p（0 < p < 1）范数的反演中.为了验证本文提出的算法的有效性,采用了三种常规模型进行模拟实验.与基于L₂范数的反演算法进行实验对比,结果表明,本文算法得到了边界清晰,磁化率分布更集中的反演结果.最后,将基于交替方向乘子算法的L₁和L_p（0 < p < 1）范数的反演应用到青海省尕林格铁矿保护区获得的实际磁异常数据中,获得了较为符合实际地质情况的稀疏反演结果.

     

A study of VLF field strength data: Both old and new

Summary Employing some old data ofRound et al. and the recent data ofHeritage et al., attenuation rates at VLF are deduced. For middle latitudes, day-time rates of less than 2 db per 1000 km of path length are not uncommon. These are in general accord with values derived from spheric waveforms and they are compatible with mode theory. An extensive bibliography is appended.     

Two-dimensional physical-based inversion of confined and unconfined aquifers under unknown boundary conditions

An inverse method is developed to simultaneously estimate multiple hydraulic conductivities, source/sink strengths, and boundary conditions, for two-dimensional confined and unconfined aquifers under non-pumping or pumping conditions. The method incorporates noisy observed data (hydraulic heads, groundwater fluxes, or well rates) at measurement locations. With a set of hybrid formulations, given sufficient measurement data, the method yields well-posed systems of equations that can be solved efficiently via nonlinear optimization. The solution is stable when measurement errors are increased. The method is successfully tested on problems with regular and irregular geometries, different heterogeneity patterns and variances (maximum K_max/K_min tested is 10,000), and error magnitudes. Under non-pumping conditions, when error-free observed data are used, the estimated conductivities and recharge rates are accurate within 8% of the true values. When data contain increasing errors, the estimated parameters become less accurate, as expected. For problems where the underlying parameter variation is unknown, equivalent conductivities and average recharge rates can be estimated. Under pumping (and/or injection) conditions, a hybrid formulation is developed to address these local source/sink effects, while different types of boundary conditions can also exert significant influences on drawdowns. Local grid refinement near wells is not needed to obtain accurate results, thus inversion is successful with coarse inverse grids, leading to high computation efficiency. Furthermore, flux measurements are not needed for the inversion to succeed; data requirement of the method is thus not much different from that of interpreting classic well tests. Finally, inversion accuracy is not sensitive to the degree of nonlinearity of the flow equations. Performance of the inverse method for confined and unconfined aquifer problems is similar in terms of the accuracy of the estimated parameters, the recovered head fields, and the solver speed.     

零偏VSP资料波阻抗反演方法研究(英文)

VSP资料钻前预测的关键在于高精度的波阻抗反演,本文针对VSP资料高分辨率、高信噪比以及能精确地分离出上、下行波的特点,提出了一种利用VSP资料进行井底以下钻头前方地层波阻抗反演的方法。该方法首先对VSP走廊叠加记录采用非线性迭代反演方法反演地下地层的波阻抗;通过在迭代过程中不断修改阻尼因子,以及引入预条件共轭梯度法求解方程组,增强了解的稳定性和收敛速度。理论模型与实际资料的处理结果表明该方法具有较好的效果,并在VSP资料钻前预测研究中具有良好的应用前景。     

Three-dimensional inversion of the single-profile magnetotelluric data

It is preferable to use the three-dimensional (3D) magnetotelluric inversion, which provides volumetric geoelectric models, to handle the array input data. However, the soundings are frequently conducted on the single profiles or on the profiles that are considerably spaced apart from each other. We explore the possibilities of the 3D inversion of such data by the example of a three-layer model containing three local inhomogeneities. We previously showed that the simple processing of the data and their 1D or 2D inversion enable reconstructing the background cross section and locating all the three inhomogeneities. In the present paper, we use this information for constructing several versions of the starting model and carrying out the smoothing 3D inversion of the data. The experiments show that if the background cross section is incorporated into the starting model, the final model provided by the inversion closely reproduces the real distribution of all geoelectric parameters. At the same time, if the starting model that hosts the inhomogeneities has the form of a homogeneous half-space, the inversion is not able to reconstruct an adequate final model.

     

Moment tensor inversion of near-source broadband data

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