Misac N. Nabighian

THE ANALYTIC SIGNAL OF TWO‐DIMENSIONAL MAGNETIC BODIES WITH POLYGONAL CROSS‐SECTION: ITS PROPERTIES AND USE FOR AUTOMATED ANOMALY INTERPRETATION

This paper presents a procedure to resoive magnetic anomalies due to two-dimensional structures. The method assumes that all causative bodies have uniform magnetization and a crosssection which can be represented by a polygon of either finite or infinite depth extent. The horizontal derivative of the field profile transforms the magnetization effect of these bodies of polygonal cross-section into the equivalent of thin magnetized sheets situated along the perimeter of the causative bodies A simple transformation in the frequency domain yields an analytic function whose real part is the horizontal derivative of the field profile and whose imaginary part is the vertical derivative of the field profile. The latter can also be recognized as the Hilbert transform of the former. The procedure yields a fast and accurate way of computing the vertical derivative from a given profile. For the case of a single sheet, the amplitude of the analytic function can be represented by a symmetrical function maximizing exactly over the top of the sheet. For the case of bodies with poiygonal cross-section, such symmetrical amplitude functions can be recognized over each corner of each polygon. Reduction to the pole, if desired, can be accomplished by a simple integration of the analytic function, without any cumbersome transformations. Narrow dikes and thin ilat sheets, of thickness less than depth, where the equivalent magnetic sheets are close together, are treated in the same fashion using the field intensity as input data, rather than the horizontal derivative. The method can be adapted straightforwardly for computer treatment. It is also shown that the analytic signal can be interpreted to represent a complex “field intensity,” derivable by differentiation from a complex “potential.” This function has simple poles at each polygon corner. Finally, the Fourier spectrum due to finite or infinite thin sheets and steps is given in the Appendix.

Toward a three-dimensional automatic interpretation of potential field data via generalized Hilbert transforms; fundamental relations

The paper extends to three dimensions (3-D) the two‐dimensional (2-D) Hilbert transform relations between potential field components. For the 3-D case, it is shown that the Hilbert transform is composed of two parts, with one part acting on the X component and one part on the Y component. As for the previously developed 2-D case, it is shown that in 3-D the vertical and horizontal derivatives are the Hilbert transforms of each other. The 2-D Cauchy‐Riemann relations between a potential function and its Hilbert transform are generalized for the 3-D case. Finally, the previously developed concept of analytic signal in 2-D can be extended to 3-D as a first step toward the development of an automatic interpretation technique for potential field data.

ADDITIONAL COMMENTS ON THE ANALYTIC SIGNAL OF TWO‐DIMENSIONAL MAGNETIC BODIES WITH POLYGONAL CROSS‐SECTION

In a previous paper (Nabighian, 1972), the concept of analytic signal of bodies of polygonal cross‐section was introduced and its applications to the interpretation of potential field data were discussed. The input data for the proposed treatment are the horizontal derivative T(x) of the field profile, whether horizontal, vertical, or total field component. As it is known, this derivative curve can be thought of as being due to thin magnetized sheets surrounding the causative bodies.

Quasi‐static transient response of a conducting half‐space— An approximate representation

For a step‐function excitation, it is shown that, to a first‐order approximation, the quasi‐static transient response in the air due to an arbitrary loop situated at the earth’s surface can be represented by a downward and outward moving current filament, of diminishing amplitude and having the same shape as the transmitter loop.

The historical development of the magnetic method in exploration

The magnetic method, perhaps the oldest of geophysical exploration techniques, blossomed after the advent of airborne surveys in World War II. With improvements in instrumentation, navigation, and platform compensation, it is now possible to map the entire crustal section at a variety of scales, from strongly magnetic basement at regional scale to weakly magnetic sedimentary contacts at local scale. Methods of data filtering, display, and interpretation have also advanced, especially with the availability of low-cost, high-performance personal computers and color raster graphics. The magnetic method is the primary exploration tool in the search for minerals. In other arenas, the magnetic method has evolved from its sole use for mapping basement structure to include a wide range of new applications, such as locating intrasedimentary faults, defining subtle lithologic contacts, mapping salt domes in weakly magnetic sediments, and better defining targets through 3D inversion. These new applications have increased the method’s utility in all realms of exploration — in the search for minerals, oil and gas, geothermal resources, and groundwater, and for a variety of other purposes such as natural hazards assessment, mapping impact structures, and engineering and environmental studies.

Historical development of the gravity method in exploration

The gravity method was the first geophysical technique to be used in oil and gas exploration. Despite being eclipsed by seismology, it has continued to be an important and sometimes crucial constraint in a number of exploration areas. In oil exploration the gravity method is particularly applicable in salt provinces, overthrust and foothills belts, underexplored basins, and targets of interest that underlie high-velocity zones. The gravity method is used frequently in mining applications to map subsurface geology and to directly calculate ore reserves for some massive sulfide orebodies. There is also a modest increase in the use of gravity techniques in specialized investigations for shallow targets. Gravimeters have undergone continuous improvement during the past 25 years, particularly in their ability to function in a dynamic environment. This and the advent of

Unification of Euler and Werner deconvolution in three dimensions via the generalized Hilbert transform

The extended Euler deconvolution algorithm is shown to be a generalization and unification of 2-D Euler deconvolution and Werner deconvolution. After recasting the extended Euler algorithm in a way that suggests a natural generalization to three dimensions, we show that the 3-D extension can be realized using generalized Hilbert transforms. The resulting algorithm is both a generalization of extended Euler deconvolution to three dimensions and a 3-D extension of Werner deconvolution. At a practical level, the new algorithm helps stabilize the Euler algorithm by providing at each point three equations rather than one. We illustrate the algorithm by explicit calculation for the potential of a vertical magnetic dipole.

The early history of the induced polarization method

This paper traces the early development of the induced polarization method, starting with field observations by Conrad Schlumberger in a mining region in France, about 1913. Starting about 1929 he introduced this technique into hydrocarbon borehole logging in the USSR, and this resulted in further development in eastern and western Europe.

Cross-hole magnetometric resistivity (MMR)

The last decade has seen a growing acceptance of the magnetometric resistivity (MMR) method as a viable exploration technique in various geologic environments. Until recently, MMR exploration was carried out with both current electrodes and recording magnetometer located on the surface of the Earth. Significant improvements in anomaly amplitude can be achieved by lowering the recording magnetometer inside a drill hole. In contrast, the lowering of current electrodes beneath the surface does not always improve surface MMR responses. The advantages of locating the magnetic detector in a drill hole are illustrated numerically, anomaly calculations being carried out with a novel yet simple integral equation technique for a plate‐like body. The practicality of the cross‐hole MMR technique is demonstrated with a successful case history. Massive sulfide mineralization is mapped at a depth exceeding 500 m.

75th Anniversary: The historical development of the magnetic method in exploration

The magnetic method, perhaps the oldest of geophysical exploration techniques, blossomed after the advent of airborne surveys in World War II. With improvements in instrumentation, navigation, and platform compensation, it is now possible to map the entire crustal section at a variety of scales, from strongly magnetic basement at regional scale to weakly magnetic sedimentary contacts at local scale. Methods of data filtering, display, and interpretation have also advanced, especially with the availability of low-cost, high-performance personal computers and color raster graphics. The magnetic method is the primary exploration tool in the search for minerals. In other arenas, the magnetic method has evolved from its sole use for mapping basement structure to include a wide range of new applications, such as locating intrasedimentary faults, defining subtle lithologic contacts, mapping salt domes in weakly magnetic sediments, and better defining targets through 3D inversion. These new applications have increased the method’s utility in all realms of exploration — in the search for minerals, oil and gas, geothermal resources, and groundwater, and for a variety of other purposes such as natural hazards assessment, mapping impact structures, and engineering and environmental studies.