DEBUBBLING: A GENERALIZED LINEAR INVERSE APPROACH*

On seismograms recorded at sea bubble pulse oscillations can present a serious problem to an interpreter. We propose a new approach, based on generalized linear inverse theory, to the solution of the debubbling problem. Under the usual assumption that a seismogram can be modelled as the convolution of the earth's impulse response and a source wavelet we show that estimation of either the wavelet or the impulse response can be formulated as a generalized linear inverse problem. This parametric approach involves solution of a system of equations by minimizing the error vector (ΔX = X^obs– X^cal) in a least squares sense. One of the most significant results is that the method enables us to control the accuracy of the solution so that it is consistent with the observational errors and/or known noise levels. The complete debubbling procedure can be described in four steps: (1) apply minimum entropy deconvolution to the observed data to obtain a deconvolved spike trace, a first approximation to the earth's response function; (2) use this trace and the observed data as input for the generalized linear inverse procedure to compute an estimated basic bubble pulse wavelet; (3) use the results of steps 1 and 2 to construct the compound source signature consisting of the primary pulse plus appropriate bubble oscillations; and (4) use the compound source signature and the observed data as input for the generalized linear inverse method to determine the estimated earth impulse response—a debubbled, deconvolved seismogram. We illustrate the applicability of the new approach with a set of synthetic seismic traces and with a set of field seismograms. A disadvantage of the procedure is that it is computationally expensive. Thus it may be more appropriate to apply the technique in cases where standard analysis techniques do not give acceptable results. In such cases the inherent advantages of the method may be exploited to provide better quality seismograms.     

CONSTRAINED INVERSION OF POTENTIAL FIELD DATA*

For constrained inversion of potential field data within the framework of generalized inversion an analysis of data error variances leads to confidence limits for the model parameters. For that purpose Pseudo-hyper-ellipsoids can be used to describe the nonlinear behaviour of the given inverse problem, and upper and lower bounds can be added to those parameters of which some independent knowledge is available. A gravity example is treated to show the application of the method.     

用于位场数据处理的广义反演技术

本文讨论了线性广义反演方法对位场延拓问题的应用。如果考虑能量有限的约束,应用拉格朗日乘子法便可得到与随机逆相同的反演公式（Franklin）,而不必假定模型为高斯白噪的随机过程。对反演算子进行谱分解之后,拉格朗日乘子起到折衷因子的作用,因此改变拉格朗日乘子的值便可找到在分辨率和误差之间取最佳折衷的反问题数值解。将这种广义反演技术用于位场从任意曲面向下延拓到源顶面,给出了较好的结果。与BG方法相比,位场向下延拓结果的精度相近,但计算速度可以提高。     

用于位场数据处理的广义反演技术

本文讨论了线性广义反演方法对位场延拓问题的应用。如果考虑能量有限的约束,应用拉格朗日乘子法便可得到与随机逆相同的反演公式（Franklin）,而不必假定模型为高斯白噪的随机过程。对反演算子进行谱分解之后,拉格朗日乘子起到折衷因子的作用,因此改变拉格朗日乘子的值便可找到在分辨率和误差之间取最佳折衷的反问题数值解。将这种广义反演技术用于位场从任意曲面向下延拓到源顶面,给出了较好的结果。与BG方法相比,位场向下延拓结果的精度相近,但计算速度可以提高。     

A SPATIAL ANALYSIS OF UPWARD CONTINUATION OF POTENTIAL FIELD DATA *

A spatial analysis of both continuous and discrete operators for upward continuation help us realize the problems and limitations which have been encountered before (Henderson 1960, Kontis 1971) but remained unsolved in practical application of upward continuation computation due to the finite length of data and operator in spatial domain. Various numerical examples show that an improvement of accuracy of continuation computations can be achieved through proper sampling and sufficient length of data.     

用改进广义逆矩阵方法解释大地电磁测深及电测深资料

本文系统阐述了广义逆矩阵反演的基本理论,着重研究解决用改进广义逆矩阵方法进行大地电磁测深和电测深曲线反演的几个具体问题,取得了良好的效果。探讨了一些规律,以求提高资料解释水平。     

用改进广义逆矩阵方法解释大地电磁测深及电测深资料

本文系统阐述了广义逆矩阵反演的基本理论,着重研究解决用改进广义逆矩阵方法进行大地电磁测深和电测深曲线反演的几个具体问题,取得了良好的效果。探讨了一些规律,以求提高资料解释水平。     

A METHOD FOR INTERPOLATION AND EXTRAPOLATION OF POTENTIAL FIELD DATA*

The application of automatic methods to the processing of potential field survey results has led to the need for the development of special procedures for the interpolation and extrapolation of such data. A method is proposed which utilizes the mathematical properties of potential field data to achieve a minimum distortion solution to the interpolation/extrapolation problem. The proposed method is suitable for use with any data point distribution. Digital computer programs have been prepared using the method; results obtained through the use of the programs are presented.     

ON THE UPWARD CONTINUATION OF POTENTIAL FIELD DATA BETWEEN IRREGULAR SURFACES*

The potential field and its derivatives at points above an irregular surface can be approximately obtained from the sampled potential field data acquired on that surface. A method of minimizing the truncation effect, which appears when gravity and magnetic maps are processed with the aid of surface integrals, is derived. The results are compared with those of the most relevant similar methods by using a theoretical, but realistic, model.     

对二维磁异常利用广义逆矩阵的选择法

本文利用广义逆矩阵理论反演解释二维磁异常。实现了加阻尼因子法,实现并改进了奇异值人工截除法;讨论了资料分辨矩阵、信息密度矩阵等辅助信息的利用;详细评价了广义逆矩阵算法的优劣。通过与目前重、磁最优化反演中常用的阻尼最小二乘法、变尺度法的对比,指出广义逆矩阵法优于这些算法。最后以河北迁安一个复杂磁异常的反演实例加以证实。     

对二维磁异常利用广义逆矩阵的选择法

本文利用广义逆矩阵理论反演解释二维磁异常。实现了加阻尼因子法,实现并改进了奇异值人工截除法;讨论了资料分辨矩阵、信息密度矩阵等辅助信息的利用;详细评价了广义逆矩阵算法的优劣。通过与目前重、磁最优化反演中常用的阻尼最小二乘法、变尺度法的对比,指出广义逆矩阵法优于这些算法。最后以河北迁安一个复杂磁异常的反演实例加以证实。     

A GENERALIZED METHOD OF COMPUTING SECOND DERIVATIVE OF GRAVITY FIELD*

The unsuitability of a particular method of computation of second derivative to varying types of field problems evinces the need of a method by which it may be possible to obtain a wide range of filters. This requirement is achieved by incorporating a mathematical smoothing operator, e^-λρ2, λ and ρ being the smoothing parameter and radial frequency respectively, in the theoretical frequency response of second derivative operation. A few sets of weights for various smoothing parameters, are presented. The analysis of Bouguer gravity anomaly map of Los Angeles Basin, California, U.S.A., is carried out by Elkins's (1951) and Rosenbach's (1953) methods and the sets of weights given in this paper which clearly establishes their filtering equivalence.     

ON THE APPLICATION OF INVERSE THEORY TO GRAVITY INTERPRETATION*

Numerical investigations have been carried out on suitable interpretation schemes for gravity, based upon Backus-Gilbert inverse theory. A two-dimensional linear model consisting of horizontal prisms (grid shape) has been adopted for interpretation of gravity profiles. An inversion technique, which uses a linear approach suitable for rapid computation, and studies with synthetic bodies have led to improvements in the method, mainly by applying weighting factors, which improve the solution in the final stages of computational work. Gravity data from the well-known Cyprus island positive anomaly were inverted to demonstrate the feasibility of the method.     

FIELD PROCESSING OF MAGNETIC DATA*

This paper deals with the practical problem of processing magnetic data in the field, and it shows how the parameters describing a geological contact can be obtained quickly and conveniently on the basis of five measurements. The complete interpretation can be performed in the field using a programmable calculator. For a small ground-party the method greatly improves the quality of the field-work and the interpretation of the magnetic data. The method has been applied to the data from a magnetic survey over the Mooki Fault (Gunnedah-Manilla, N.S.W.), and the results are discussed as an illustration of the efficacity of the method as a field procedure.     

CURVED RAYPATH INTERPRETATION OF SEISMIC REFRACTION DATA*

Vertical velocity gradients can be readily accommodated in seismic refraction interpretations by using simple mathematical idealizations of curved raypaths entailed by certain analytic representations of these gradients. Computation procedures are formulated for velocity gradients in the overburden or an intermediate layer of a multilayer section. The former is demonstrated for an overburden velocity gradient manifest in published time-distance data.     

A NEW APPROACH TO MAGNETIC PROFILE INTERPRETATION*

By using two components of anomalous magnetic fields and a formulation including complex numbers it is possible to calculate the position parameters of thick plates and both magnetization and position of thin plates directly from any two or three points of anomaly profiles. The formulae (interpretation operators) allow automatic topographic corrections to be made. The new two-component operators give more reliable results than the conventional methods of interpretation. The variance of the parameter values obtained with subsequent points of an anomaly measures directly, the total error of interpretation. The application of infinite thin plate operators to a long profile results in characteristic patterns, from which the estimation of the number of plates and their approximate position is possible.     

A NEW APPROACH TO THE INVERSION OF SEISMIC DATA*

A new inversion method for seismic reflection data is based on impedance concepts and uses transformations to reduce the essentially non-linear problem to a linear eigenvalue formulation without approximation. A set of reflection data has been inverted using this method. The characteristic impedances of the layers so determined are compared with log data from a reference borehole.     

A TIME-DOMAIN APPROACH TO THE NORMAL-INCIDENCE INVERSE PROBLEM*

This tutorial paper presents a time-domain derivation of the Normal equations which are associated with the normal-incidence inverse problem. Connections are made between our time-domain derivation and the more familiar z-transform transfer function derivation.     

DIRECT INTERPRETATION OF RESISTIVITY DATA OVER TWO-DIMENSIONAL STRUCTURES*

A direct interpretation scheme is developed which is capable of determining most of the geological features of a ground which can be assumed to be two dimensional in structure. This scheme extends the earlier work of Pekeris (1940) and Koefoed (1968) to the case where the basal layer of a ground is undulating. It also has a limited use for finding the parameters of a dipping dyke in the lower medium. Though the top and dip of the dyke can be determined, this is not true for the thickness.