L. B. Slichter

The Theory of the Interpretation of Seismic Travel‐Time Curves in Horizontal Structures

The theory of the interpretation of seismic travel‐time curves for refracted rays in horizontal structures is treated after the manner of Herglotz‐Wiechert, under the customary assumption that the ray paths obey the laws of geometrical optics. Multiple valued travel‐time curves, discontinuous velocity functions, and the discontinuous travel‐time curves associated with a slower speed bed receive special consideration. It appears that interpretations satisfactory from the theoretical point of view may be obtained in these cases, although, experimentally, sufficiently complete data to meet the requirements of theory may often be difficult or impossible to obtain.

The Interpretation of the Resistivity Prospecting Method for Horizontal Structures

The direct problem of deducing from electrical potentials observed at the surface of a horizontally uniform earth the unknown variation of the conductivity with depth reduces to a boundary value problem of unusual type. Its solution for the isotropic case is developed in Part I, following. In Part II the more usual inverse problem is solved for some special classes of conductivity functions and graphical examples are given as an aid in guiding the interpretation work. In Part III, the anisotropic case is discussed.

Cooling of the earth

In Part I observational data and hypotheses pertinent to the cooling problem of the earth are reviewed. The possible importance of heat transport by slow thermal convection currents is indicated. In Part II the theory of the cooling of a solid radioactive earth is presented by a simple method which identifies this problem with an equivalent one in the cooling of a nonradioactive earth. The facts as to slowness of cooling for temperature perturbations of long wave length are reviewed. The importance of the large heat capacity of the earth is emphasized by means of simple examples. If the earth is cooling the rate must be very small, but, if heating occurs, higher rates are possible. Thermal considerations indicate that solidification of the mantle probably occurred from the bottom up. Because the steady-state temperatures in a radioactive earth are significant in determining whether the earth is heating or cooling at depth, such temperature distributions have been computed for a dozen or more types of radioactive earths. It is probable that thermal equilibrium in the earth as a whole is far from achieved. The earth may be heating everywhere—cooling everywhere—or heating at some depths and cooling at others. The transient temperatures are expressed, and the transient heat flows computed for a number of types of suitable radioactive earths which are amenable to convenient computation. The case of heat flow in layered earths is especially examined. The observed surface heat flows fix definite upper limits to the total heat generation which may exist within about 100 or 200 kilometers of the surface. It is important to improve the reliability and scope of present information concerning (1) surface heat flows, (2) the structure and composition of the crust, and (3) the association of heat generation with crustal layers. Until better knowledge of all three of these factors in the same region is available, estimates of the amount of radioactivity existing in and below the crust will involve large uncertainties. Even granting excellent thermal information about the crust, it is possible to deduce by thermal theory only very limited information about regions below a depth of 200 or 300 km. Significant concentrations of radioactivity may exist all the way to the center without violating known criteria. At depth the problem of the cooling of the earth is nonunique to such a degree that little information about the deep interior may be deduced from purely thermal data.

An Inverse Boundary Value Problem in Electrodynamics

From knowledge of the electromagnetic field at the surface of a half‐space due to a prescribed oscillatory source, is determined the unknown variation with depth of the conductivity and the dielectric constant. Unique solutions exist for these quantities.

FIELD OF AN ALTERNATING MAGNETIC DIPOLE ON THE SURFACE OF A LAYERED EARTH

The magnetic field near a vertical alternating magnetic dipole on the surface of a layered earth is computed for points on the surface. In the layer the dimensionless conductivity parameter is assumed to take the values 0, 14, 1, and 4; in the homogeneous substratum, this parameter is assigned the values 0, 14, 1, 4, and infinity. The induced field is computed at distances from the source 12, 1, 2, 4, 8, and 16 times the layer thickness.

EARTH’S FREE MODES AND A NEW GRAVIMETER

Gravitational observations of free vibrations of the earth due to the Alaskan earthquake of March, 1964 were compared with corresponding results for the Chilean earthquake. In the graver spheroidal modes of periods greater than 7 minutes which were compared, the periods generally agreed within a few parts per thousand. The freemode spectrum has been used by Anderson, Landisman, and others to improve models of the earth. To fit both the free mode periods and the seismic data, Anderson’s modification of classical earth models introduces a more rapid increase of density in the upper mantle, and less rapid increase near the core. In addition to the precise set of numbers characterizing the free periods, the free mode observations supply another independent set of criteria in the form of the “Q” values for the different modes. The large Q value of more than 25,000 for the purely radial mode excited by the Alaskan quake is noteworthy. In gravity observations of the earth’s oscillations, of period one hour or mo...

Seismic interpretation theory for an elastic earth

The seismic interpretation problem for an isotropic spherical earth is analyzed on the basis of elastic theory, under the assumption that the three independent elastic parameters are unknown continuous functions of the depth. It is shown that solutions for these functions may be obtained in the form of Taylor’s series. The problem is treated for three types of symmetrical excitation conditions on the free surface: (1) a shear source of type prϕ only; (2) a pressure distribution with vanishing surface shear stress; (3) an excitation consisting of pressure in combination with surface shear stress of type prθ. In each case the excitation functions are arbitrary functions of time. It is assumed that the associated components of surface displacement over the sphere are known from available observations, as functions of time. Thus, the complete information contained in seismic records is used in the proposed interpretation process, without need of selecting, identifying and assigning arrival times to specific events on the records. The two static elastic parameters may theoretically be determined from observations at a single frequency, including the frequency zero, or static case. The determination of the dynamic elastic parameter requires the use of at least two frequencies. Algebraic checks are obtained by comparing the general solutions with the corresponding results for two special cases in which the elastic parameters vary in a prescribed manner in the interior of the sphere. In both these cases treatment by the classical ray-path method of interpretation is excluded, because the wave velocity decreases with depth. Furthermore, the ray-path method (which is essentially a method of geometrical optics) would fail to distinguish between the two examples in any case, since the velocity function is the same in both, although the elastic parameters differ. In contrast to the valuable ray-path method, the analytical procedures in the present solution of the elastic problem are prohibitively cumbersome. Practical application of elastic theory to the direct interpretation of seismograms requires further development of the theory with probable utilization of modern high-speed computing methods.

AN ELECTROMAGNETIC INTERPRETATION PROBLEM IN GEOPHYSICS

An interpretation problem in electromagnetic prospecting is discussed. A flat earth in which the three electrical properties of material vary only with depth is subjected to an alternating inducing field produced by a dipole above the surface with axis perpendicular to the surface. Observations of the horizontal or of the vertical component of the magnetic intensity at the ground’s surface are supposed to be available at all distances. From these observations solutions for the three unknown functions are developed. When the magnetic permeability is variable, the solutions for the permeability and dielectric functions require observations at two different frequencies. The conductivity function may be found from observations at a single frequency. It is shown that the horizontal and vertical components of the magnetic field intensity are mutually dependent in the region above the ground’s surface; and formulae independent of the ground’s characteristics are deduced for expressing Hρ(ρ,zi) in terms of Hz(ρ,z...

An Electromagnetic Interpretation Problem for the Sphere

Within an isotropic sphere, the three electrical properties of material are assumed to vary in an unknown manner as functions of the radial distance only. The sphere is subjected to a symmetrical harmonic inducing field from an external source. From observations of the magnetic vector on the surface of the sphere, the variation of its electrical properties is determined in the form of Taylor’s series. The variations of the magnetic permeability and dielectric constant are determined from observations at two frequencies; the conductivity function, however, may be found by use of a single frequency. The solutions are checked algebraically by comparison with known results for two special cases. It is shown that the horizontal and vertical components of the magnetic vector are mutually dependent. A simple reciprocal relationship which is formally independent of the electrical properties of the sphere exists between the distribution of the magnetic intensity and that of the source currents. The converse problem of determining the unknown electrical properties of the exterior region when those of the sphere are known is also discussed. This problem is an idealized version of the problem of determining the electrical properties of the high atmosphere.