RESISTIVITY SOUNDING ON A MULTI-LAYERED EARTH WITH TRANSITIONAL LAYERS. PART I: THEORY*

The electrical potential generated by a point source of current on the ground surface is studied for a multi-layered earth formed by layers alternatively characterized by a constant conductivity value and by conductivity varying linearly with depth. The problem is accounted for by solving a Laplace's differential equation for the uniform layers and a Poisson's differential equation for the transitional layers. Then, by a simple algorithm and by the introduction of a suitable kernel function, the general expression of the apparent resistivity for a Schlumberger array placed on the surface is obtained. Moreover some details are given for the solution of particular cases as 1) the presence of a infinitely resistive basement, 2) the absence of any one or more uniform layers, and 3) the absence of any one or more transitional layers. The new theory proves to be rather general, as it includes that for uniform layers with sharp boundaries as a particular case. Some mathematical properties of the kernel function are studied in view of the application of a direct system of quantitative interpretation. Two steps are considered for the solution of the direct problem: (i) The determination of the kernel function from the field measurements of the apparent resistivity. Owing to the identical mathematical formalism of the old with this new resistivity theory, the procedures there developed for the execution of the first step are here as well applicable without any change. Thus, some graphical and numerical procedures, already published, are recalled. (ii) The determination of the layer distribution from the kernel function. A recurrent procedure is proposed and studied in detail. This recurrent procedure follows the principle of the reduction to a lower boundary plane, as originally suggested by Koefoed for the old geoelectrical theory. Here the method differs mainly for the presence of reduction coefficients, which must be calculated each time when passing to a reduced earth section.     

Solutions of the heat conduction equation in a non-uniform soil

A class of analytic, periodic solutions of the heat conduction equation in a non-uniform soil is derived. The class may be characterized by the fact that the speed of the temperature wave varies according to the square root of the soil diffusivity (a function of soil depth). In addition it is shown that the constant soil solution is the limiting case when the rate of change with depth of diffusivity and thermal conductivity become very small. The solutions may be regarded as general whenever temperature analysis is restricted to small values of depth or whenever the soil parameters vary slowly. For all other cases the class of solutions possess the additional property that the rate of change of conductive capacity varies directly as the product of the bulk density and specific heat of the soil. A particular temperature profile is given for the case when the diffusivity varies as the nth power of depth.     

RESISTIVITY SOUNDING ON AN EARTH MODEL CONTAINING TRANSITION LAYERS WITH LINEAR CHANGE OF RESISTIVITY WITH DEPTH*

Mallick and Roy solved the problem of determining the apparent resistivity function for a three-layer stratification in which the central layer is a transition layer with linear change of the conductivity with depth. In the present paper the problem is solved for a transition layer with linear change of the resistivity with depth, a type of change that seems to be more common in nature than the type considered by Mallick and Roy. The solution is extended to layer stratifications involving an arbitrary number of transition layers and of homogeneous layers. The solution is given in the form of a modification of the recurrence relation that was derived by Pekeris for homogeneous layers.     

Magnetotelluric and seismological determination of the lithosphere-asthenosphere transition in Central Europe

We assembled all available magnetotelluric (MT) sounding curves that cover characteristic provinces of Europe. The depths of layers with increased electrical conductivities estimated from MT sounding curves by the respective authors were compared with a model of the lithosphere thickness derived from seismic P residuals. Regions of a relatively thin and thick lithosphere are characterized by positive and negative P residuals, respectively.

We have found a good correspondence with both sets of results. In the central part of the Pannonian Basin the depth of the lithosphere-asthenosphere transition is estimated at 60–80 km and 100–120 km at the flanks. The results for the Bohemian Massif show deeper levels of this transition between 100 and 200 km. These depths seem to be typical of the lithosphere base beneath the Variscan domain of Central Europe.     

Calibration of base flow separation methods with streamflow conductivity

The conductivity mass-balance (CMB) method can be used to calibrate analytical base flow separation methods. The principal CMB assumptions are base flow conductivity is equal to streamflow conductivity at lowest flows, runoff conductivity is equal to streamflow conductivity at highest flows, and base flow and runoff conductivities are assumed to be constants over the period of record. To test the CMB assumptions, fluid conductivities of ground water, surface runoff, and streamflow were measured during wet and dry conditions in a 12-km(2) stream basin. Ground water conductivities at wells varied an average of 6% from dry to wet conditions, while stream conductivities varied 58%. Shallow ground water conductivity varied significantly with distance from the stream, with lowest conductivities of 87 microS/cm near the divide, a maximum of 520 microS/cm 59 m from the stream, and 215 microS/cm 22 m from the stream. Runoff conductivities measured in three rain events remained nearly constant, with lower conductivities of 35 microS/cm near the divide and 50 microS/cm near the stream. The CMB method was applied to the records from 10 USGS stream-gauging stations in Texas, Kentucky, Georgia, and Florida to calibrate the USGS base flow separation technique, HYSEP, by varying the time parameter 2N*. There is a statistically significant relationship between basin areas and calibrated values of 2N*, expressed as N = 0.46A(0.44), with N in days and A in km(2). The widely accepted relationship N = 0.83A(0.2) is not valid for these basins. Other analytic methods can also be calibrated with the CMB method.     

Sharp boundary inversion of 2D magnetotelluric data

We consider 2D earth models consisting of laterally variable layers. Boundaries between layers are described by their depths at a set of nodes and interpolated laterally between nodes. Conductivity within each layer is described by values at a set of nodes fixed within each layer, and is interpolated laterally within each layer. Within the set of possible models of this sort, we iteratively invert magnetotelluric data for models minimizing the lateral roughness of the layer boundaries, and the lateral roughness of conductivities within layers, for a given level of data misfit. This stabilizes the inverse problem and avoids superfluous detail. This approach allows the determination of boundary positions between geological units with sharp discontinuities in properties across boundaries, while sharing the stability features of recent smooth conductivity distribution inversions.
We compare sharp boundary inversion results with smooth conductivity distribution inversion results on a numerical example, and on inversion of field data from the Columbia River flood basalts of Washington State. In the synthetic example, where true positions and resistivities are known, sharp boundary inversion results determine both layer boundary locations and layer resistivities accurately. In inversion of Columbia flood basalt data, sharp boundary inversion recovers a model with substantially less internal variation within units, and less ambiguity in both the depth to base of the basalts and depth to resistive basement.     

华北地区上地幔及过渡带电性结构研究

采用远参考道和Robust技术,处理了华北地区14个地磁台站资料,得到了相干度超过0.8的地磁测深响应函数.并将其转换为大地电磁测深的响应函数,获取了105~107 s周期范围内的视电阻率和相位.应用ρ+理论对数据进行了一致性检验和反演,结果表明417km,850km深度附近可能存在电性间断面.同时采用基于一维最光滑模型的Occam反演方法得到了300~1000km范围的地幔电性结构,并与前人在其他地区的研究结果进行了对比.发现华北地区地幔过渡带的电导率在大兴安岭—太行山重力梯度带东西两侧表现不同,重力梯度带附近及西侧台站下方过渡带深度的电导率和北美的Tucson地区相当,而华北地区东部的电导率在地幔过渡带范围高出西侧约2~5倍,这很可能和太平洋板块的俯冲有关.     

混合物整体电导率的研究

研究由几种导体成分掺杂的混合物的整体电导率.对欧姆定律求平均,得到混合物电导率定义.对电流连续性方程求平均,得到混合物中电场增量方程.求电场增量方程在同种成分上的平均,并结合混合物电导率定义,得到混合物电导率公式.现有的三种混合物结构下电导率公式（电导率串联公式、并联公式和整体各向同性混合物电导率公式）都是混合物电导率公式的特例.进一步分析得出结论,混合物整体电导率是各成分电导率与整体电导率结构并联后的体积串联.     

Horizontal magnetic dipole over an inhomogeneous earth model with exponential variation of conductivity

The paper outlines the formulation of the problem of calculating the electromagnetic field components due to a horizontal magnetic dipole placed over a multilayered earth model with one of the layers having exponential variation of conductivity with depth. Analytical solutions and numerical computations are performed for three-layered earth models possessing the described conductivity variation in the transition layer. It is assumed that the conduction currents dominate the displacement currents. Results presented here show the influence of the transition layer thickness and the conductivity contrast between the top and the bottom layers on electric and magnetic field components. The results show a characteristic dependence on the conductivity inhomogeneity.     

RESISTIVITY SOUNDING ON A MULTILAYERED EARTH CONTAINING TRANSITION LAYERS*

For curves over a horizontally stratified earth where any of the layers has a conductivity variation proportional to (1 +β·z)^N, where β and N are arbitrary constants and z is the depth to the layer, expressions for apparent resistivity for Wenner and Schlumberger sounding are derived. No assumption has been made about the continuity of conductivity at the interface. It is shown that most of the previous investigations in this connection can be regarded as particular cases of the present study.     

混合物电导率和介电常数的研究

应用电场理论研究混合物的电性，得出了混合物的电导率公式和介电常数公式．这一结果表明：混合物的电导率决定于混合物各成分电导率的均值和方差；混合物的介电常数决定于混合物各成分介电常数的均值和方差．作为所得结果的应用，给出了岩石的电导率公式和介电常数公式．     

超高压下氧化物电导率测量及下地幔电导率的讨论

在金刚石对顶砧压机上,对一些氧化物进行了0—30 GPa 压力范围下的电导率测量,发现所测氧化物可按其电导性质为两类:过渡族元素氧化物和非过渡族元素氧化物前者平均电导率较高且有正压效应,后者平均电导率较低且压力效应不明显.过渡族元素氧化物的高导性质对讨论下地幔的电导事和化学模式具有重要意义.结合前人的研究成果,半定量地估计了700 km 处下地幔的电导率.     

APPARENT RESISTIVITY OF A MULTILAYERED EARTH WITH A LAYER HAVING EXPONENTIALITY VARYING CONDUCTIVITY*

If the conductivity of any one of the layers of a horizontally stratified earth varies exponentially with depth with or without a discontinuity at the interface, the corresponding expressions for apparent resistivity for Wenner- and Schlumberger-sounding arrays can be formulated. The general case has been broadly divided into three categories for mathematical simplicity. All previous discussions of this problem can be regarded as particular cases of the present study.     

MODAL ANALYSIS OF SOFT-SOIL SITES INCLUDING RADIATION DAMPING

For the one-dimensional analysis of soft-soil layers on an elastic half-space, a general form of analytical solution is developed for converting radiation damping due to energy leaking back to the half-space into equivalent modal damping, allowing the modal analysis technique to be extended to a site where radiation damping has to be accounted for. Closed-form solutions for equivalent modal damping ratios and effective modal participation factors are developed for a single layer with a shear wave velocity distribution varying from constant to linearly increasing with depth. Compact and recursive forms of solutions for equivalent modal damping ratios are developed for a system with an arbitrary number of homogeneous layers on an elastic half-space. Comparisons with numerical solutions show that the modal solutions are accurate. The nominal frequency of a site, i.e. the inverse of four times the total shear wave travel time through the layers, is an important parameter for estimating the high mode frequencies. A parameter study shows that for the same impedance ratio of the bottom layer to the elastic half-space, a system of soil layers with an increasing soil rigidity with depth has, in general, larger peak modal amplifications at the ground surface than does a single homogeneous layer on an elastic half-space, while a system with a decreasing soil rigidity with depth has smaller modal peak amplifications. © 1997 by John Wiley & Sons, Ltd.     

Electrical conductivities of pyrolitic mantle and MORB materials up to the lowermost mantle conditions

The electrical conductivities of natural pyrolitic mantle and MORB materials were measured at high pressure and temperature covering the entire lower mantle conditions up to 133 GPa and 2650 K. In contrast to the previous laboratory-based models, our data demonstrate that the conductivity of pyrolite does not increase monotonically but varies dramatically with depth in the lower mantle; it drops due to high-spin to low-spin transition of iron in both perovskite and ferropericlase in the mid-lower mantle and increases sharply across the perovskite to post-perovskite phase transition at the D″ layer. We also found that the MORB exhibits much higher conductivity than pyrolite. The depth–conductivity profile measured for pyrolite does not match the geomagnetic field data below about 1500-km depth, possibly suggesting the existence of large quantities of subducted MORB crust in the deep lower mantle. The observations of geomagnetic jerks suggest that the electrical conductivity may be laterally heterogeneous in the lowermost mantle with high anomaly underneath Africa and the Pacific, the same regions as large low shear-wave velocity provinces. Such conductivity and shear-wave speed anomalies are also possibly caused by the deep subduction and accumulation of dense MORB crust above the core–mantle boundary.     

Regional induction studies: A review of methods and results

The geomagnetic skin-effect is specified by setting three length scales in relation to each other: L₁ for the overhead source. L₂ for the lateral non-uniformity of the subsurface conductor, L₃ for the depth of penetration of a quasi-uniform transient field into this conductor. Relations for the skin-effect of a quasi-uniform source in layered conductors are generalized to include sources of any given geometry by introducing response kernels as functions of frequency and distance. They show that only those non-uniformities of the source which occur within a distance comparable to L₃ from the point of observation are significant. The skin-effect of a quasi-uniform source in a laterally non-uniform earth is expressed by linear transfer functions for the surface impedance and the surface ratio of vertical/horizontal magnetic variations. In the case of elongated structures and E-polarisation of the source, a modified apparent resistivity is defined which as a function of depth and distance gives a first orientation about the internal distribution of conductivity. The skin-effect of a non-uniform source in a non-uniform earth is considered for stationary and “running” sources. Recent observations on the sea floor and on islands indicate a deep-seated change of conductivity at the continent—ocean transition, bringing high conductivity close to the surface, a feature which may not prevail, however, over the full width of the ocean. There is increasingly reliable evidence for high conductivities (0.02 to 0.1 micro ^?1 m^?1) at subcrustal or even at crustal depth beneath certain parts of the continents, in some cases without obvious correlation to geological structure.     

Love wave dispersion: A comparison of results for a semi-infinite medium with inhomogeneous layers and for its approximation by homogeneous layers

Love wave dispersion in various semi-infinite media consisting of inhomogeneous layers is discussed. The phase and group velocities are computed when shear wave velocity and density in each inhomogeneous layer are varying exponentially with depth. At the beginning one or two inhomogeneous layers over a homogeneous semi-infinite medium are considered. The dispersion results for these structures are compared with those for their approximations with homogeneous layers. Comparisons show that differences of phase and group velocities for the original models from those for their approximated models (i) increase with the increase of wave number and (ii) are larger for group velocity than for phase velocity. The difference is approximately proportional to the rate of change of parameters in the layers. Finally, dispersion curves are obtained for model IP3MC, which consists of many inhomogeneous and homogeneous layers over a homogeneous semi-infinite medium. The results are compared with the observed group velocity data across the Indian Peninsula.     

Amplitudes of long-period PcP and the core-mantle boundary

The amplitudes of the core reflection PcP are sensitive to the wave velocities and densities in the neighborhood of the core-mantle boundary (CMB). We study the amplitude ratio of the long-period phases PcP and P from two South American deep-focus earthquakes with favorable fault-plane solution, depth and magnitude, as recorded by WWNSS and CSN stations in North America.Comparison is made with long-period PcP/P amplitude ratios, derived from theoretical seismograms for a variety of CMB models. Models from previous studies, which were mainly derived from short-period PcP observations and which are characterized by discrete layers above the CMB, are almost all inconsistent with the long-period data. The data also discriminate against low nonzero S velocities below the CMB. Simple first-order-discontinuity models of the CMB, for instance according to the Jeffreys-Bullen earth model or according to recent models based mainly on free oscillations, explain the data reasonably well.Model improvements are attempted by varying the P-velocity gradient above the CMB. The best amplitude fit is obtained for a rather strong decrease in P velocity with depth in this zone which, however, gives no acceptable traveltime fit for PcP. The scatter in body-wave amplitudes is considerable even for long-period waves and may prevent the correct assessment of that part of the amplitude variation of a phase with distance that is due to the variation of velocities and densities with depth alone.     

LOCATING A DIFFRACTOR BELOW PLANE LAYERS OF CONSTANT INTERVAL VELOCITY AND VARYING DIP*

For a two-dimensional situation, a direct solution exists to the problem of locating a point diffractor below plane layers of constant interval velocity and varying dip. The required surface measurements are obtainable from an arbitrary portion of the observed diffraction curve on the stacked section. The method resembles the computing of plane dipping layers of constant interval velocity from common depth point surface measurements.     

电导率随深度分层线性变化的大地电磁响应

本文在电导率随深度分层线性变化的地球模型上,研究了大地电磁的正演问题.由Airy微分方程导出了电场和磁场水平分量的解、地面阻抗的递推表达式和视电阻率的计算公式;并证明了分层线性模型与均匀层状模型一样,也存在高阻薄层的H等值性和低阻薄层的S等值性;总结了电导率随深度的变化率α,对大地电磁测深曲线的影响规律.