The Computation of Second Vertical Derivatives of Geomagnetic Fields†

Published:January 01, 1967

CiteCitation
Roland G. Henderson, Isidore Zietz, 1967. "The Computation of Second Vertical Derivatives of Geomagnetic Fields", Mining Geophysics Volume II, Theory, Don A. Hansen, Walter E. Heinrichs, Jr., Ralph C. Holmer, Robert E. MacDougall, George R. Rogers, John S. Sumner, Stanley H. Ward
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Abstract
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.
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Contents
Mining Geophysics Volume II, Theory
The relative merits of any geophysical method in a given situation can be predicted by careful study of the expected messagetonoise1 ratio. For example, let us draw or deduce from the subsequent text, the anomaly formulas due to a spherical inhomogeneity in the subsurface and the symbols in each formula are explained in the text. The gravity, magnetic, resistivity, and inducedpolarization surveys all are volume dependent, whereas the electromagnetic method is dependent only upon the area of the inhomogeneity, normal to the inducing field. Thus, a thin disk can give nearly the same electromagnetic anomaly as a sphere of the same radius.
If we can make a reasonable estimate of the physical property contrast anticipated to exist between ore and host, we can then predict the anomaly magnitude expected from the sphere, when buried at any given depth, via the geometric factor. Note that from this viewpoint, given the maximum or saturation value of unity for the physical property factor, the magnetic and resistivity methods theoretically give the same percent anomaly due to a sphere. The physical property function M–iN for the electromagnetic method has a maximum value of one half for a sphere while the change with frequency of the electrical resistivity contrast.
Thus, except for a factor of two, the magnetic, resistivity, electromagnetic, and inducedpolarization methods should give the same maximum anomaly. Note that the geometry of the anomalous fields for each of these methods is an induced dipole with a resultant falloff of peak anomaly proportional to the inverse cube of the depth to the center of the sphere below the measuring plane. In contrast, the gravity method exhibits an inverse second power falloff due to an induced monopole. The density contrast between ore and host sometimes exhibits a maximum value of two. Thus f r om a maximum message viewpoint, one would be inclined to rate the methods in the order given above. However, we need to counter this bias by considering expected values of the physical property factor and the noise for any given geologic situation.
Let us look, then, at iron ore, massive sulfides, and disseminated sulfides, items treated i n Volume I. We should expect the following physical property ranges: A very wide range of properties is evident and hence the prediction of an anomaly magnitude looks hopeless.