Generalized Laplace coefficients and Newcomb derivatives

It is shown how the generalized Laplace coefficients can be employed to deduce explicit formulas for ordinary and Newcomb derivatives of the Laplace coefficients.      

Interpretation and modelling of regional gravity data of the Aveiro Basin (Northwest Portugal)

The gravity map of the Aveiro Basin, North West Portugal was produced in 2000/2001 and some limited two dimensional (2D) interpretation was carried out. It is intended in this article to advance the interpretation and modelling of the original Aveiro basin gravity data. Thus, the data were processed and the horizontal gradient, the second vertical derivative and downward continued field computed. The corresponding maps have been processed and a new interpretation of the basin tectonic features has been proposed. This is compared with the previous geological and tectonic information obtained from former surveys and local boreholes. As the next step a more detailed modelling of the Aveiro Basin took place. Bearing the interpretation of the basin tectonic features in mind, several north–south and east–west gravity profiles were established and modelled. The proposed models were presented and discussed. As a result, a comprehensive mass distribution model for the basin was proposed. Finally, comparison was made between the new gravity model and the previously published tectonic features map. This modelling is particularly useful for groundwater protection, exploration and exploitation and also for the dimensioning of drilling operations. Bearing in mind the geomorphology of the basin the gravity method is a very economical tool for the investigation of basin structures.     

扩展的倾斜角和倾斜角总水平导数方法

为了突出显示重磁场中地质体分布、断裂构造特征等信息,使图像显示错落有序、层次分明、清楚直观,有利于解释人员从不同角度认识重磁场异常特征.本文通过分析倾斜角概念,根据反余切函数特性,提出了扩展的倾斜角(iTdr)和倾斜角总水平导数(iTdr_Thdr)方法,来增强重磁异常信息,突出异常形状特点,以便于识别重磁异常中地质体边界和提取地质体分布、断裂构造平面展布特征等信息.理论模型和实际资料对比分析表明该方法简单、实用、具有较好的识别效果和较高的分辨能力,能够获取丰富的地质信息,对识别地质体边界、划分大地构造单元、确定断裂带和地质构造走向等具有实际意义.     

A direct method for interpreting magnetic anomalies—the case of a horizontal circular cylinder

A new method is introduced here to interpret the magnetic anomalies with special reference to vertical magnetic anomalies of a horizontal circular cylinder. The parameters of the cylinder are found to be related toV, ∂V/∂x and∂V/∂z at the origin of the cylinder. HereV is the observed anomaly,∂V/∂x and∂V/∂z are its horizontal and vertical derivatives respectively. The origin may be located working out an equality,viz.,3(∂V/∂z) ²=2V∂²V/∂z² which is true only at the origin. Thus, once the cylinder is located,V, ∂V/∂x and∂V/∂z at its origin can be determined and hence its parameters. The procedure is illustrated with a theoretical example.     

Automatic differentiation in geophysical inverse problems

Automatic differentiation (AD) is the technique whereby output variables of a computer code evaluating any complicated function (e.g. the solution to a differential equation) can be differentiated with respect to the input variables. Often AD tools take the form of source to source translators and produce computer code without the need for deriving and hand coding of explicit mathematical formulae by the user. The power of AD lies in the fact that it combines the generality of finite difference techniques and the accuracy and efficiency of analytical derivatives, while at the same time eliminating 'human' coding errors. It also provides the possibility of accurate, efficient derivative calculation from complex 'forward' codes where no analytical derivatives are possible and finite difference techniques are too cumbersome. AD is already having a major impact in areas such as optimization, meteorology and oceanography. Similarly it has considerable potential for use in non-linear inverse problems in geophysics where linearization is desirable, or for sensitivity analysis of large numerical simulation codes, for example, wave propagation and geodynamic modelling. At present, however, AD tools appear to be little used in the geosciences. Here we report on experiments using a state of the art AD tool to perform source to source code translation in a range of geoscience problems. These include calculating derivatives for Gibbs free energy minimization, seismic receiver function inversion, and seismic ray tracing. Issues of accuracy and efficiency are discussed.     

Magnetic interpretation of dyke anomalies using derivatives

The horizontal and vertical derivative profiles of magnetic anomalies of dykes show some interesting properties. The points of zero derivatives and the points where the derivatives are equal are conjugate point pairs. A method of interpretation of dyke anomalies is suggested, which utilizes the distances between these points.Notation F Magnetic anomaly in total intensity - Z Depth to top of the dyke - 2T Width of the dyke - Geological dip of the dyke - I Effective intensity of magnetisation in the plane of profile - Dip of effective magnetisation vector in the plane of profile - Strike angle of the dyke - i Magnetic dip - Q – - Q _f –+arctan (sin coti) - I _f      

The gravitational potential and its derivatives for the prism

 As a simple building block, the right rectangular parallelepiped (prism) has an important role mostly in local gravity field modelling studies when the so called flat-Earth approximation is sufficient. Its primary (methodological) advantage follows from the simplicity of the rigorous and consistent analytical forms describing the different gravitation-related quantities. The analytical forms provide numerical values for these quantities which satisfy the functional connections existing between these quantities at the level of numerical precision applied. Closed expressions for the gravitational potential of the prism and its derivatives (up to the third order) are listed for easy reference. Received: 18 August 1999 / Accepted: 15 June 2000