A Least-squares Approach to Depth Determination from Numerical Horizontal Self-potential Gradients

We have developed a least-squares minimization approach to depth determination of a buried ore deposit from numerical horizontal gradients obtained from self-potential (SP) data using filters of successive window lengths (graticule spacings). The problem of depth determination from SP gradients has been transformed into the problem of finding a solution to a nonlinear equation of the form f(z)=0. Formulas have been derived for vertical and horizontal cylinders and spheres. Procedures are also formulated to estimate the electrical dipole moment and the polarization angle. The method is applied to synthetic data with and without random noise. Finally, the validity of the method is tested on two field examples. In both cases, the depth obtained is found to be in a very good agreement with that obtained from drilling information.     

Fair Function Minimization for Direct Interpretation of Residual Gravity Anomaly Profiles Due to Spheres and Cylinders

A new interpretative approach is proposed to interpret residual gravity anomaly profiles in order to determine the depth, the amplitude coefficient and the geometric shape factor of simple spherical and cylindrical buried structures. This new approach is based on both Fair function minimization and on stochastic optimization modeling. The validity of this interpretative approach is demonstrated through studying and analyzing two synthetic gravity anomalies, using simulated data generated from a known model with different random noises components and a known statistical distribution. Being theoretically proven, this new approach has been applied on three real field gravity anomalies from Sweden, Senegal and the United States. The agreement between the results obtained by the proposed method and those obtained by other interpretation methods is good and comparable.     

A Least-squares Minimization Approach to Depth Determination from Magnetic Data

—We have developed a least-squares minimization approach to depth determination from magnetic data. By defining the anomaly value T(0) at the origin and the anomaly value T(N) at any other distance (N) on the profile, the problem of depth determination from magnetic data has been transformed into finding a solution to a nonlinear equation of the form f(z)=0. Formulas have been derived for a sphere, horizontal cylinder, dike, and for a geologic contact. Procedures are also formulated to estimate the effective magnetization intensity and the effective magnetization inclination. A scheme for analyzing the magnetic data has been formulated for determining the model parameters of the causative sources. The method is applied to synthetic data with and without random errors. Finally, the method is applied to two field examples from Canada and Arizona. In all cases examined, the estimated depths are found to be in goodagreement with actual values.     

A Least-squares Approach to Shape Determination from Residual Self-potential Anomalies

—We have developed a least-squares minimization approach to determine the shape (shape-factor) of a buried polarized body from a residual self-potential anomaly profile. By defining the zero anomaly distance and the anomaly value at the origin on the profile, the problem of the shape-factor determination is transformed into the problem of finding a solution of a nonlinear equation of the form f(q) = 0. Procedures are also formulated to estimate the depth of polarization angle, and the electric dipole moment. The method is applied to synthetic data with and without random noise. The obtained shape-factor agrees very well with the model shape-factor when using synthetic data. After adding ± 2 percent random error in the synthetic data, the shape factor obtained is within ± 4 percent. Finally the validity of the method is tested on a field example from the Ergani copper district, Turkey.     

An Automatic Method of Direct Interpretation of Residual Gravity Anomaly Profiles due to Spheres and Cylinders

We have developed a least-squares minimization approach to determine the depth and the amplitude coefficient of a buried structure from residual gravity anomaly profile. This approach is basically based on application of Werner deconvolution method to gravity formulas due to spheres and cylinders, and solving a set of algebraic linear equations to estimate the two-model parameters. The validity of this new method is demonstrated through studying and analyzing two synthetic gravity anomalies, using simulated data generated from a known model with different random error components and a known statistical distribution. After being theoretically proven, this approach was applied on two real field gravity anomalies from Cuba and Sweden. The agreement between the results obtained by the proposed method and those obtained by other interpretation methods is good and comparable. Moreover, the depth obtained by the proposed approach is found to be in very good agreement with that obtained from drilling information.     

基于重力异常反演的三维地震波速度结构分析方法研究

利用重力异常反演测试三维地震波速度结构,存在解不唯一、可靠性不高的问题。将面波反演充分融合到重力异常反演方程中,降低传统反演方法的非唯一性,并提升可靠性。以川滇地区为例,采用融合后的重力异常反演方法分析三维地震波速度结构。通过速度和密度的关系转换,得到对应的重力异常数据。由于面波频射数据主要对地震波横波速度敏感,因此将重力异常数据和初始横波速度相连,依据地震波速度和岩石密度之间的关系,获取重力异常反演方程,用于分析速度结构。选取21.6°~34.2°N、97.1°~105.9°E范围内的川滇地区活动块体作为实验数据,经过实验分析发现：使用该方法迭代反演川滇地区地壳上地幔顶部横波速度,重力异常数据和面波频射数据的残差值分别是6.24 mGal和0.027 km/s,实际拟合效果较好;分析该地区不同深度切面横波速度发现,在24 km深度处,上地壳中含有相对低速层,在44 km深度处,中下地壳中存在低速层;且该方法分析川滇地区三维地震波速度结构解的分辨率较高。     

A Versatile Nonlinear Inversion to Interpret Gravity Anomaly Caused by a Simple Geometrical Structure

A geophysical interpretative method is proposed to depth, amplitude coefficient and geometrical shape factor determination of a buried structure from an observed gravity anomaly related to a cylinder or a sphere-like structure.The method is based on nonlinearly constrained mathematical modelling and also on stochastic optimization approaches. The proposed interpretative method first has been tested on theoretical synthetic models with different random errors at a certain depth, where a very close agreement has been observed between assumed and evaluated parameters. Subsequent field data have been considered for which the interpreted results by other methods are available for comparison. The agreement between the obtained results by the proposed technique and by other geophysical methods is good. A statistical analysis has been also carried out to demonstrate the accuracy and the precision of the suggested interpretative method.     

A New Algorithm for Depth Determination from Total Magnetic Anomalies due to Spheres

We have developed an automatic method to determine the depth of a buried sphere from numerical second horizontal derivative anomalies obtained from total field magnetic data. The method is based on using a relationship between the depth and a combination of observations at symmetric points with respect to the coordinate of the projection of the center of the source in the plane of the measurement points with a free parameter (graticule spacing). The problem of depth determination has been transformed into the problem of finding a solution of a nonlinear equation of f(z) = 0. Procedures are also formulated to determine the magnetic moment and the effective angle of magnetization. The method is applied to synthetic examples with and without random errors and tested on a field example from Senegal. In all cases, the depth solutions are in good agreement with the actual ones.     

Determination of Sedimentary Basin Basement Depth: A Space Domain Based Gravity Inversion using Exponential Density Function

An automatic inversion using ridge regression algorithm is developed in the space domain to analyze the gravity anomalies of sedimentary basins, among which the density contrast decreases with depth following a prescribed exponential function. A stack of vertical prisms having equal widths, whose depths become the unknown parameters to be estimated, describes the geometry of a sedimentary basin above the basement complex. Because no closed form analytical equation can be derivable in the space domain using the exponential density-depth function, a combination of analytical and numerical approaches is used to realize forward gravity modeling. The depth estimates of sediment-basement interface are initiated and subsequently improved iteratively by minimizing the objective function between the observed and modeled gravity anomalies within the specified convergence criteria. Two gravity anomaly profiles, one synthetic and a real, are interpreted using the proposed technique to demonstrate its applicability.     

Depth Estimation from the Scaling Power Spectral Density of Nonstationary Gravity Profile

-- A technique to estimate the depth to anomalous sources from the scaling power spectra of long nonstationary gravity profiles is presented. The nonstationary profile is divided into piecewise stationary segments based on the criterion of optimum gate length in which the time-varying and time-invariant autocorrelation functions are similar. The division of a nonstationary into piecewise stationary allows identification of the portion of the crust with different geological histories, and using the stationary portion of the gravity profiles, more consistent depths to the anomalous sources have been obtained. The technique is tested with the synthetic gravity profile and applied along the Jaipur-Raipur geotransect in western and central India. The geotransect has been divided into four stationary parts: Vindhyan low, Bundelkhand low, Narmada rift and Chhattisgarh basin; each section corresponding to a different geological formation. Forward modeling of gravity data using results of each stationary section is carried out to propose the subsurface structure along the Jaipur-Raipur transect.     

基于融合重力异常的大别造山带东段莫霍面反演研究

利用实测布格异常和EGM2008重力异常融合结果,采用频率域位场反演方法计算大别造山带东段莫霍面三维空间分布,结合区域地质构造和地震活动等资料讨论大别造山带东段莫霍面分布特征及其构造含义等。研究结果显示:①莫霍面空间分布体现了块体构造差异,大别造山带莫霍面最深,最大深度达42km,显示东大别造山带存在明显山根,扬子地块深度次之,华北地块最浅;郯庐断裂带及其东侧区域存在莫霍面上隆带;②大别造山带北侧和南侧莫霍面陡变带分别位于青山—晓天断裂附近和襄樊—广济断裂以北,体现华北地块和扬子地块向大别造山带之下俯冲的构造特征,指示了深部构造缝合带位置;莫霍面深度特征表明郯庐断裂带区域构造边界带属性明显,其切割深度至少达到壳幔边界,大别造山带商城—麻城断裂两侧经历了差异隆升;③研究区域绝大多数地震发生在莫霍面以上深度,莫霍面深度陡变带、上隆带及不同莫霍面深度特征区的转换带为区域地震活动提供了深部构造条件。     

GPS Height and Gravity Variations Due to Ocean Tidal Loading Around Taiwan

This study presents predicts ocean tidal loading (OTL) effects using a Green’s function approach and validates a novel tidal model for Taiwan. Numerical integration of OTL is performed using the Gauss quadrature method and a local tidal model for the inner zone and a global model for the outer zone. Observed time series of GPS-derived vertical displacements and gravity variations (3–7 days) at five co-located GPS-gravimeter stations along the South East China and Taiwan coasts were utilized to assess the accuracy of the proposed models and two other models. The OTL-induced gravity variations are 3–16 μgal and the vertical site displacements are 13–27 mm. Generally, an OTL model using a mixed global and local tidal model generates better agreement with the observations than an OTL model using a global tidal model only. However, containing a local model inside a global model does not always produce a good agreement with the observations. The relatively large discrepancies between modeled and observed OTL values at some stations indicate that there is a need for an improved local tidal model in the study area. An erratum to this article can be found at     

Optimal Model for Geoid Determination from Airborne Gravity

Two different approaches for transformation of airborne gravity disturbances, derived from gravity observations at low-elevation flying platforms, into geoidal undulations are formulated, tested and discussed in this contribution. Their mathematical models are based on Green's integral equations. They are in these two approaches defined at two different levels and also applied in a mutually reversed order. While one of these approaches corresponds to the classical method commonly applied in processing of ground gravity data, the other approach represents a new method for processing of gravity data in geoid determination that is unique to airborne gravimetry. Although theoretically equivalent in the continuous sense, both approaches are tested numerically for possible numerical advantages, especially due to the inverse of discretized Fredholm's integral equation of the first kind applied on different data. High-frequency synthetic gravity data burdened by the 2-mGal random noise, that are expected from current airborne gravity systems, are used for numerical testing. The results show that both approaches can deliver for the given data a comparable cm-level accuracy of the geoidal undulations. The new approach has, however, significantly higher computational efficiency. It would be thus recommended for real life geoid computations. Additional errors related to regularization of gravity data and the geoid, and to accuracy of the reference field, that would further deteriorate the quality of estimated geoidal undulations, are not considered in this study.     

An Easy Method for Interpretation of Gravity Anomalies Due to Vertical Finite Lines

A new method is introduced to determine the top and bottom depth of a vertical line using gravity anomalies. For this, gravity at a distance x from the origin and horizontal derivative at that point are utilized. A numerical value is obtained dividing the gravity at point x by horizontal derivative. Then a new equation is obtained dividing the theoretical gravity equation by the derivative equation. In that equation, assigning various values to the depth and length of vertical line, several new numerical values are obtained. Among these values, a curve is obtained for the one that is closest to the first value from attending the depth and length values. The intersection point of these curves obtained by repeating this procedure several times for different points x yield the real depth and length values of the line. The method is tested on two synthetics and field examples. Successful results are obtained in both applications.     

Depth Estimation of Simple Causative Sources from Gravity Gradient Tensor Invariants and Vertical Component

The gravity gradient tensor (GGT) is deduced from products of second-order derivatives of the gravitational potential. A new method based on the invariants of the GGT has been proposed in this research to interpret gravity data due to sphere, infinite horizontal cylinder and semi-infinite vertical cylinder. The method estimates the depth of these simple causative sources from the multiplication of the maximum of the gravity vertical component by the maximum value of the invariants I ₁ to I ₂ ratio. To show the reliability and correctness of the estimated depths on 3-D models, the method has been tested using theoretical data with and without random noise. In addition, I have applied the method to a field-data example in Texas, USA and the depth obtained by the present method is compared with those published in the literature.     

An Approach for Passive Removal of Residual LNAPL from Groundwater

Light nonaqueous phase liquids (LNAPLs) are a problematic challenge for obtaining site closure or no further action remediation sites. The source of the LNAPLs varies from leaking underground petroleum storage tanks, to manufacturing facilities where oil leaks create LNAPL accumulations beneath factory floors. Active recovery using pumping or periodic vacuum recovery from wells or sumps is used for remediation, but usually has disappointing results when LNAPL reaccumulates to thicknesses exceeding the 0.01-foot action level recognized by many states. This paper presents a simple passive approach for recovering persistent LNAPL using nonwoven hydrophobic oil absorbing cloth. The method used laboratory trials to assess physical properties of the cloth. Parameters observed and assessed included sorptive capacity and rate, buoyancy, and LNAPL wicking. It was determined that the cloth could be rolled and secured with cable ties for placement in the wells/sumps. Two placement designs were developed, one where rolled sorbent freely floated on the well/sump fluid surface and a second where the sorbent roll was placed in the fluid column at a fixed depth. Sorbents were then used at two manufacturing facilities where LNAPLs persisted for decades. In both instances, many wells/sumps were reduced to thicknesses below the action level in less than 2 months. In most wells, LNAPL did not reaccumulate. Where it did reaccumulate, it was less than 50% of the original thickness. Using laboratory-derived recovery rates, cloth sorbents could be sized to minimize placement/recovery frequency while effectively recovering LNAPL.