Three-Dimensional Gravity Modeling In All Space

We review available analytical algorithms for the gravity effect and gravity gradients especially the vertical gravity gradient due to a right rectangular prism, a right polygonal prism, and a polyhedron. The emphasis is placed on an investigation of validity, consistency, and especially singularities of different algorithms, which have been traditionally proposed for calculation of the gravity effect on ground (or outside anomalous bodies), when they are applied to all points in space. The rounding error due to the computer floating point precision is estimated. The gravity effect and vertical gradient of gravity in three dimensions caused by a cubic model are calculated by different types of algorithms. The reliability of algorithms for the calculation of gravity of a right polygonal prism and a polyhedron is further verified by using a regular polygonal prism approximating a vertical cylinder and a regular polyhedron approximating a sphere, respectively. By highlighting Haáz-Jung-Plouff and Okabe-Steiner-Zilahi-Sebess' formulae for a right rectangular prism, Plouff's algorithm for a right polygonal prism, and Gouml;tze and Lahmeyer's algorithm for a polyhedron and removing their singularities, we demonstrate that these formulae and algorithms can be used to model the gravity anomaly and its vertical gradient at all possible computation positions.     

The gravity effect of a homogeneous polyhedron for three-dimensional interpretation

Summary As an aid to the interpretation of gravity data for three-dimensional causative bodies, a method of evaluation of the gravity effect of a homogeneous polyhedron at an external point has been worked out. It has been shown that the gravity effect is expressible in terms of some prinitive integrals over the basic triangular faces of the polyhedron. The formula for evaluation of these integrals has also been derived.Contribution from the Earth Physics Branch No. 501.     

Petrogenetic grid for ultrahigh-pressure metamorphism in the model system CaO-MgO-SiO2-CO2-H2O

Abstract Petrogenetic grids for ultrahigh-pressure (UHP) metamorphism were calculated at different X_co2 conditions in the model system CaO-MgO-SiO₂-CO₂-H₂O involving coesite (Co), diopside (Di), dolomite (Do), enstatite (En), forsterite (Fo), magnesite (Ms), quartz (Qz), talc (Tc), tremolite (Tr) using a published internally consistent thermodynamic data set. Two P-T grids at X_co2= 0.01 and 0.5 are described. In the calculated P-T grid at X_co2= 0.01, four out of 10 stable invariant points, Co-En-Ms-Tc, Co-Di-En-Tc-Tr, Co-Di-Ms-Tc-Tr and Di-En-Ms-Tc-Tr lie within the stability field of coesite. If the fluid phase has X_co2= 0.5, no invariant point is stable under UHP conditions. Some magnesite-bearing assemblages are stabilized by the following three reactions: Di + Ms = Do + Fo + CO₂, Ms + Tr = Do + Fo + CO₂+ H₂O and Ms + Tc = Fo+ CO₂+ H₂O at X_co2= 0.01 and by reaction Ms + Tc = Fo + CO₂+ H₂O together with these three at X_co2= 0.5. Ten possible UHP assemblages for mafic and ultramafic compositions at very low X_co2 conditions include the following: Co-Do-Ms, Co-Di-Ms, Co-Di-Tc, Di-Ms-Tc, Di-En-Tc-, Di-En-Ms, Co-Di-En, Di-En-Fo, Di-Fo-Ms, Di-Do-Fo. Among them, talc-bearing assemblages are restricted to X_co2 < 0.02 and their high-P limit is 31.7 kb (749°C) at X_co2= 0.01. Dolomite-magnesite-silica assemblages have large P-T stability fields even if X_co2 is as low as 0.1, and could occur in cold subduction zones with very low geothermal gradients. Reported UHP coesite-dolomite assemblage is restricted only to a calc-silicate rock interlayered with marble where X_co2 is relatively higher; no such assemblage appears for mafic and ultramafic rocks with low X_co2 evidenced by the occurrence of diopside (or omphacite) at the expense of dolomite + coesite. The effect of X_co2 on the stability of coesite-dolomite-magnesite, diopside-enstatite-magnesite, diopside-talc assemblages is examined and the occurrence of coesite-dolomite, magnesite-bearing and talc-bearing assemblages in the Dabie UHP rocks are interpreted by employing the calculated P-T grids.     

基于MPICH环境下的复杂重力异常体并行正演模拟

任意多面体重力异常正演公式常用于解决复杂几何形态地质体的正演问题。 本文以均匀物性多面体重力异常正演公式为基础, 应用有限元技术中的网格离散化思想, 以任意四面体为基本单元, 通过并行计算技术在MPICH环境下实现了任意连续空间物性分布复杂异常体网格模型的重力异常正演模拟, 通过并行处理可以有效加速正演计算速度。 本文研究结果对于联合重力异常场正演建模和开展复杂模型网格的重力场计算有一定参考意义。     

Effective calculation of gravity effects of uniform triangle polyhedra

Uniform tetrahedra are commonly used elementary bodies for gravity calculations from which arbitrary polyhedra can be composed. A simple derivation of the gravity effect is presented for the apex P of the tetrahedron expanded from P to an arbitrarily oriented plane triangle. Integration of its potential effect in a rotated coordinate system applies vector algebra and renders the anomalous potential depending on the distance of P over the triangle plain and a function of the triangle coordinates. Partial differentiation by moving P infinitesimally in z-direction leads to two terms, a simple and a complex one; they can be understood as describing the same difference from two points of view: leaving P at the apex of the changed polyhedron or moving P off the unchanged polyhedron. Both views imply the same shape change and the sum over the polyhedron is thus numerically equal. Hence we need to calculate only the one of the terms of the differential which is simpler. The calculation of the gravity effect is numerically simplified and more stable. This has been tested for many models and is demonstrated by two examples.     

The Field of Attraction of a Polyhedron and Polygonal Plate with Linear Density

New representations of the elements of the fields of attraction (potential and its first derivatives) are presented for the important approximating models such as polyhedron and polygonal plates with the density varying by the linear law. It is shown that these elements are determined through the elements of the fields from the models with the known analytical representations (polyhedron, polygonal plate, and material segment with constant densities) and the additional integrals for which the explicit analytical expressions exist.     

A Geostatistically Based Ground Water Monitoring Study of Nonpoint Source No3--N Concentrations

Geostatistical interpretations of ground water monitoring data are presented to define the spatial distributions of NO₃^--N in the ground water at two demonstration test sites in the Idaho Snake River Plain. Sequential Gaussian simulation was used to delineate monthly ground water NO₃^--N changes during and after implementation of a prescribed crop rotation at test site 1. Trend surface analyses were used to illustrate monthly ground water NO₃^--N changes during and after a prescribed irrigation practice was implemented at test site 2. These evaluations suggest that geostatistically based ground water monitoring can be effective in the delineation of changes in ground water quality in shallow, unconfined aquifers in agricultural areas such as those in southern Idaho. Geostatistical methods showed spatial and temporal changes in ground water NO₃^--N inferred to be a result of the agricultural practices implemented.     

230Th/234U and 14C dating of a lowstand coral reef beneath the insular shelf off Irabu Island, Ryukyus, southwestern Japan

Abstract   High-resolution seismic reflection profiles delineated the distribution of mound-shaped reflections, which were interpreted as reefs, beneath the insular shelf western off Irabu Island, Ryukyus, southwestern Japan. A sediment core through one of the mounded structures was recovered from the sea floor at a depth of −118.2 m by offshore drilling and was dated by radiometric methods. The lithology and coral fauna of the core indicate that the mounded structure was composed of coral–algal boundstone suggesting a small-scaled coral reef. High-precision α-spectrometric ²³⁰Th/²³⁴U dating coupled with calibrated accelerator mass spectrometric ¹⁴C ages of corals obtained reliable ages of this reef ranging from 22.18 ± 0.63 to 30.47 ± 0.98 ka. This proves that such a submerged reef was formed during the lowstand stage of marine oxygen isotope stages 3–2. The existence of low-Mg calcite in the aragonitic coral skeleton of 22.18 ± 0.63 ka provides evidence that the reef had once been exposed by lowering of the relative sealevel to at least −126 m during the last glacial maximum in the study area. There is no room for doubt that a coral reef grew during the last glacial period on the shelf off Irabu Island of Ryukyus in the subtropical region of western Pacific.     

Computing three-dimensional gravitational fields with equivalent sources

Existing techniques for computing the gravitational field due to a homogeneous polyhedron all transform the required volume integral, expressing the field due to a volume distribution of mass, into a surface integral, expressing the potential due to a surface mass distribution over the boundary of the source body. An alternative representation is also possible and results in a surface integral expressing the potential due to a variable-strength double layer located on the polyhedral source boundary. Manipulation of this integral ultimately allows the gravitational field component in an arbitrary direction to be expressed as a weighted sum of the potentials due to two basic source distributions. These are a uniform-strength double layer located on all faces and a uniform-strength line source located along all edges. The derivatives of the gravitational field components can also be expressed in a similar form as can the magnetic field components due to a homogeneous magnetic polyhedron. It follows that the present approach can be used to generate a universal program capable of modelling all the commonly used potential field responses due to 3D bodies of arbitrary shape.     

Characterization of dipping fractures in a transverselyisotropic background

Although it is believed that natural fracture sets predominantly have near‐vertical orientation, oblique stresses and some other mechanisms may tilt fractures away from the vertical. Here, we examine an effective medium produced by a single system of obliquely dipping rotationally invariant fractures embedded in a transversely isotropic with a vertical symmetry axis (VTI) background rock. This model is monoclinic with a vertical symmetry plane that coincides with the dip plane of the fractures. Multicomponent seismic data acquired over such a medium possess several distinct features that make it possible to estimate the fracture orientation. For example, the vertically propagating fast shear wave (and the fast converted PS‐wave) is typically polarized in the direction of the fracture strike. The normal‐moveout (NMO) ellipses of horizontal reflection events are co‐orientated with the dip and strike directions of the fractures, which provides an independent estimate of the fracture azimuth. However, the polarization vector of the slow shear wave at vertical incidence does not lie in the horizontal plane – an unusual phenomenon that can be used to evaluate fracture dip. Also, for oblique fractures the shear‐wave splitting coefficient at vertical incidence becomes dependent on fracture infill (saturation). A complete medium‐characterization procedure includes estimating the fracture compliances and orientation (dip and azimuth), as well as the Thomsen parameters of the VTI background. We demonstrate that both the fracture and background parameters can be obtained from multicomponent wide‐azimuth data using the vertical velocities and NMO ellipses of PP‐waves and two split SS‐waves (or the traveltimes of PS‐waves) reflected from horizontal interfaces. Numerical tests corroborate the accuracy and stability of the inversion algorithm based on the exact expressions for the vertical and NMO velocities.     

Tectonothermal evolution of the Lesser Himalaya, Nepal: Constraints from 40Ar/39Ar ages from the Kathmandu Nappe

Abstract The Himalaya is a fold-and-thrust wedge formed along the northern margin of the Indian continent, and consists of three thrust-bounded lithotectonic units; the Sub-Himalaya, the Lesser Himalaya, and the Higher Himalaya with the overlying Tethys Himalaya from south to north, respectively. The orogen-scale, intracrustal thrusts which bound the above lithotectonic units are splays off an underlying subhorizontal dkcollement, and show a southward propagating piggy-back sequence with an out-of-sequence thrust. Among these thrusts, the Main Central Thrust zone (MCT zone) has played a major role in Himalayan tectonics. The MCT zone represents a shear zone which has accommodated southward thrusting of the Higher Himalayan crystalline thrust sheet over the Lesser Himalayan sequence for ～140 km. The Kathmandu Nappe in central Nepal has been transported over the Lesser Himalayan metasediments along the MCT zone, and is locally separated from the Higher Himalayan thrust sheet in the north by an out-of-sequence thrust. ⁴⁰Ar/³⁹Ar ages have been determined for one whole-rock phyllite and six muscovite concentrates from metasedimenta-ry rocks and variably deformed granites in the Kathmandu Nappe. These ages range from 44 Ma to 14 Ma, and suggest a record of both Eo-Himalayan (Eocene) and Neo-Himalayan (Miocene) tectonothermal events in the Tertiary Himalayan orogeny. The Miocene event was associated with translation along the MCT zone. No tectonothermal event of the Late Miocene to Early Pliocene ages have been reported near the MCT zone in southern Lesser Himalayan crystalline nappe or klippe, although such events have been documented within and around the MCT zone in the northern root zone of the Higher Himalaya. This suggests that out-of-sequence thrusting may have occurred between 14 Ma and 5 Ma, probably during the period 10-7.5 Ma. Since then the frontal MCT zone below the Kathmandu Nappe has been inactive, but the MCT zone in the northern root zone has remained active. The rapid increase in denudation rates of the Higher Himalaya since the Late Miocene may have been caused by ramping along the out-of-sequence thrust at depth.     

Optimum expression for computation of the magnetic field of a homogeneous polyhedral body12

An expression which is optimum with respect to the simplicity of the numerical computations is obtained for the magnetic field of a polyhedron with constant magnetization. The high accuracy of the results is illustrated using a realistic numerical model. The existence of the magnetic field at points inside the source and on its boundary is discussed and related to real magnetic data modelling.     

Adaptive subtraction of multiples using the L1-norm

A strategy for multiple removal consists of estimating a model of the multiples and then adaptively subtracting this model from the data by estimating shaping filters. A possible and efficient way of computing these filters is by minimizing the difference or misfit between the input data and the filtered multiples in a least‐squares sense. Therefore, the signal is assumed to have minimum energy and to be orthogonal to the noise. Some problems arise when these conditions are not met. For instance, for strong primaries with weak multiples, we might fit the multiple model to the signal (primaries) and not to the noise (multiples). Consequently, when the signal does not exhibit minimum energy, we propose using the L₁‐norm, as opposed to the L₂‐norm, for the filter estimation step. This choice comes from the well‐known fact that the L₁‐norm is robust to ‘large’ amplitude differences when measuring data misfit. The L₁‐norm is approximated by a hybrid L₁/L₂‐norm minimized with an iteratively reweighted least‐squares (IRLS) method. The hybrid norm is obtained by applying a simple weight to the data residual. This technique is an excellent approximation to the L₁‐norm. We illustrate our method with synthetic and field data where internal multiples are attenuated. We show that the L₁‐norm leads to much improved attenuation of the multiples when the minimum energy assumption is violated. In particular, the multiple model is fitted to the multiples in the data only, while preserving the primaries.