Roland G. Henderson

Iterative three-dimensional solution of gravity anomaly data using a digital computer

Given gridded gravity anomaly values and certain limiting restrictions on the mass distribution, a three‐dimensional structural model can be calculated automatically from gravity anomaly data by successive approximations. The causative body is optionally assumed to be either flat‐topped, flat‐bottomed, or symmetrical about a horizontal plane. Density contrast and the position of a horizontal plane delimiting either the top, base, or midsection of the causative body must be specified. A first approximation of structure is obtained by means of the Bouguer slab relationship. The gravity field of this first model is calculated and at each grid point the ratio of observed to calculated gravity is used to modify the first structural model, thus leading to a second approximation of structure. The process is iterated until a satisfactory agreement between observed and calculated gravity is obtained.

THE COMPUTATION OF SECOND VERTICAL DERIVATIVES OF GEOMAGNETIC FIELDS

Second vertical derivatives of magnetic fields, because of their high resolving power, are often very useful in interpreting magnetic anomalies. Formulas are developed which permit their ready numerical computation. Comparisons are made between the resulting approximate values and the rigorous values obtained for simple idealized fields. The similarity between maps of second vertical derivatives of fields and those of certain types of residual fields is discussed.

A COMPREHENSIVE SYSTEM OF AUTOMATIC COMPUTATION IN MAGNETIC AND GRAVITY INTERPRETATION

In the interpretation of magnetic and gravity anomalies, downward continuation of fields and calculation of first and second vertical derivatives of fields have been recognized as effective means for bringing into focus the latent diagnostic features of the data. A comprehensive system has been devised for the calculation of any or all of these derived fields on modern electronic digital computing equipment. The integral for analytic continuation above the plane is used with a Lagrange extrapolation polynomial to derive a general determinantal expression from which the field at depth and the various derivatives on the surface and at depth can be obtained. It is shown that the general formula includes as special cases some of the formulas appearing in the literature. The process involves a “once for all depths” summing of grid values on a system of concentric circles about each point followed by application of the appropriate one or more of the 19 sets of coefficients derived for the purpose. Theoretical a...

The upward continuation of anomalies in total magnetic intensity fields

In airborne magnetometry consideration must be given to the necessity of multi-level observations. The problem of computing total intensity anomalies from data observed on lower levels is investigated in the case of contours exhibiting both two- and three-dimensional features. It is found that even fairly complex anomalies can be extended, with results differing but slightly from observations at the higher level. Maps for comparing computed and observed anomalies are presented. The mathematical basis is given together with numerical formulas and procedures for affecting the computations.

Reduction Of Unevenly Spaced Potential Field Data To A Horizontal Plane By Means Of Finite Harmonic Series

Conventional reductions of gravity and magnetic data do not lead to values that are effectively on the same horizontal plane, although it is common practice to regard them so. In regions of high topographic relief, failure to take into account local differences in vertical gradients can result in appreciable error. In this study a method is developed for reducing to a common level gravity or magnetic anomaly data observed at unevenly spaced stations at various elevations above a reference plane. The reduction is effected by means of finite harmonic series approximations in which the coefficients are determined by matrix methods and least squares. Traditional Fourier methods are not applicable because uneven station spacing and relative vertical displacement of stations preclude the use of the orthogonality properties of the trigonometric functions. The number of terms required to represent the data adequately is discussed in terms of “cutoff” wavenumbers empirically determined from residual variance estim...

ANALYSIS OF TOTAL MAGNETIC‐INTENSITY ANOMALIES PRODUCED BY POINT AND LINE SOURCES

The component of total magnetic intensity in the direction of the earth’s total field, the quantity measured by the airborne magnetometer, is studied in relation to point‐pole and line‐of‐poles sources. Theoretical profiles are examined for maxima and minima, and it is established that the depth is a linear function of the half‐maximum abscissa. A family of curves is presented by means of which factors can be obtained for use in estimating depths. The well‐known factors used with vertical intensity profiles are included as a special case. An example is given in which the factors are used in analyzing a theoretical anomaly.

On The Validity Of The Use Of The Upward Continuation Integral For Total Magnetic Intensity Data

The integral solving the Dirichlet problem for a plane, known as the “upward continuation integral” to exploration geophysicists working with magnetic and gravity fields, is sometimes misunderstood. Recently, some geophysicists have commented that its use in continuing ΔT, the component of the total intensity anomaly in the direction of the earth’s normal field, is suspect, the thought being that only components normal to the surface can be so continued. The integral in question is ΔT(x,y,z)=-z2π⋅∫-∞∞∫-∞∞ΔT(α,β)dαdβ[(x-α)2+(y-β)2+z2]32,where ΔT(α, β) represents measured total intensity values on the plane of observations z=0, in a right handed rectangular coordinate system in which the z axis is positive vertically downward.

A PRELIMINARY REPORT ON MODEL STUDIES OF MAGNETIC ANOMALIES OF THREE‐DIMENSIONAL BODIES

Model experiments were made to devise a rapid method for calculating magnetic anomalies of three‐dimensional structures. The magnetic fields of the models were determined using the equipment at the Naval Ordnance Laboratory, White Oaks, Md. An irregularly shaped mass was approximated by an array of prismatic rectangular slabs of constant thickness and varying horizontal dimensions. Contoured maps are being prepared for these magnetic models at different depths and for several magnetic inclinations. The fields of these three‐dimensional structures are obtained by super‐imposing the appropriate contoured maps and adding numerically the effects at each point. The equipment and laboratory methods are described. Theoretical and practical examples are given.

Graphical calculation of total-intensity anomalies of three-dimensional bodies

Calculation of the total‐intensity anomaly of a three‐dimensional body of arbitrary shape is greatly facilitated by the orthographic projection of a topographic map of the body onto a plane normal to the inducing field. The graphical integration is then effected by a modified Gassmann integration process. Applications to theoretical and laboratory models establish the relative accuracy of the method. Examples are given of applications to observed anomalies over two laccoliths, Round Butte and Square Butte, in Montana.

The Sudbury [Ontario] aeromagnetic map as a test of interpretation methods

The method of interpretation of aeromagnetic maps described by Vacquier, Steenland, Henderson, and Zietz (1951) is applied to several anomalies on the aeromagnetic map of part of Sudbury, Ontario. The resulting average computed depths deviate from the known depths by less than 10 percent. The aeromagnetic map is described and compared with the known surface geology.