Lindrith Cordell

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.

Limitations of determining density or magnetic boundaries from the horizontal gradient of gravity or pseudogravity data

The horizontal‐gradient method has been used since 1982 to locate density or magnetic boundaries from gravity data (Cordell, 1979) or pseudogravity data (Cordell and Grauch, 1985). The method is based on the principle that a near‐vertical, fault‐like boundary produces a gravity anomaly whose horizontal gradient is largest directly over the top edge of the boundary. Magnetic data can be transformed to pseudogravity data using Fourier techniques (e.g., Hildenbrand, 1983) so that they behave like gravity data; thus the horizontal gradient of pseudogravity also has maximum magnitude directly over the boundary. The method normally is applied to gridded data rather than to profiles. The horizontal‐gradient magnitude is contoured and lines are drawn or calculated (Blakely and Simpson, 1986) along the contour ridges. These lines presumably mark the top edges of magnetic or density boundaries. However, horizontal‐gradient magnitude maxima (gradient maxima) can be offset from a position directly over the boundary f...

Gravity analysis using an exponential density-depth function; San Jacinto Graben, California

In the analysis of gravity data over thick sedimentary basins, density contrast can sometimes be approximated by a continuous function decreasing exponentially with depth. Typical values of the exponential decrement are in the order of 0.3 to 1.5 km (super -1) . The gravity effect of an infinite (Bouguer) slab in this system tends to a finite limit as the slab becomes infinitely thick, leading to quantitative, and sometimes stringent, limitations on the resolving power of gravity analysis over deep structures. The exponential density term acts as an integrating factor in deriving the gravity effect of a prismatic building element, leading to a simple expression involving both the gravity field and its vertical derivative. A recursive computer algorithm giving the inverse solution automatically in terms of assigned density parameters is applied to a gravity profile over a graben structure at San Jacinto, California. The computed basin configuration is consistent with the seismically determined basement depth of 2.4 km and the estimated density layering of the graben fill.

A terracing operator for physical property mapping with potential field data

The terracing operator works iteratively on gravity or magnetic data, using the sense of the measured fields local curvature, to produce a field comprised of uniform domains separated by abrupt domain boundaries. The result is crudely proportional to a physical-property function defined in one (profile case) or two (map case) horizontal dimensions. This result can be extended to a physical-property model if its behavior in the third (vertical) dimension is defined, either arbitrarily or on the basis of the local geologic situation. The terracing algorithm is computationally fast and appropriate to use with very large digital data sets. Where gravity and magnetic data are both available, terracing provides an effective means by which the two data sets can be compared directly.Results of the terracing operation somewhat resemble those of conventional susceptibility (or density) mapping. In contrast with conventional susceptibility mapping, however, the terraced function is a true step function, which cannot be depicted by means of contour lines. Magnetic or gravity fields calculated from the physical-property model do not, in general, produce an exact fit to the observed data. By intent, the terraced map is more closely analogous to a geologic map in that domains are separated by hard-edged domain boundaries and minor within-domain variation is neglected.The terracing operator was applied separately to aeromagnetic and gravity data from a 136 km X 123 km area in eastern Kansas. Results provide a reasonably good physical representation of both the gravity and the aeromagnetic data. Superposition of the results from the two data sets shows many areas of agreement that can be referenced to geologic features within the buried Precambrian crystalline basement. The emerging picture of basement geology is much better resolved than that obtained either from the scanty available drill data or from interpretation of the geophysical data by inspection.

A scattered equivalent-source method for interpolation and gridding of potential-field data in three dimensions

Potential‐field geophysical data observed at scattered discrete points in three dimensions can be interpolated (gridded, for example, onto a level surface) by relating the point data to a continuous function of equivalent discrete point sources. The function used here is the inverse‐distance Newtonian potential. The sources, located beneath some of the data points at a depth proportional to distance to the nearest neighboring data point, are determined iteratively. Areas of no data are filled by minimum curvature. For two‐dimensional (2-D) data (all data points at the same elevation), grids calculated by minimum curvature and by equivalent sources are similar, but the equivalent‐source method can be tuned to reduce aliasing. Gravity data in an area of high topographic relief in southwest U.S.A. were gridded by minimum curvature (a 2-D algorithm) and also by equivalent sources (3-D). The minimum‐curvature grid shows strong correlation with topography, as expected, because variation in gravity effect due to...

The decompensative gravity anomaly and deep structure of the region of the Rio Grande Rift

An isostatic correction is commonly made to Bouguer anomaly gravity data to remove the gravity effect of isostatic compensation of topographic loads. In the USSR a “decompensative” correction has then been made to the isostatic gravity anomaly to remove the gravity effect of isostatic compensation of geologic loads as well. Under an hypothesis of local (as opposed to regional) compensation the gravity effect of a shallow geological body can be separated from that of its inferred deep compensating root by deconvolution. By contrast with the procedure originally defined by Zorin, we employ here calculations in the wave number domain, leading to an efficient and exact solution. The decompensative transfer function acts to increase amplitude of the isostatic anomaly at all wavelengths, but especially at the low end of the isostatic anomaly spectrum in the 200–600 km wavelength region, where anomaly amplitudes may increase by as much as a factor of 2. In a 1200×1200 km region centered on the Rio Grande rift the decompensative correction ranges from about −35 to +25 mGal. The rift is not particularly apparent in the decompensative correction nor is the rift any more apparent in the decompensative anomaly than in the isostatic anomaly. The decompensative anomaly, however, highlights an arcuate gravity low and a system of gravity highs inferred to reflect prerift welts of mass concentration which have indirectly influenced the position of the rift and its segmentation and zones of accommodation. Under the assumptions made, if the decompensative anomaly is subtracted from the Bouguer anomaly, then the residual is the gravity anomaly field of deep structure, without gravity effects of shallow sources in the upper crust. We suppose this gravity field to be dominated by the gravity effects of the density contrast across the crust-mantle boundary (Moho) and the density contrast at the lithosphere-asthenosphere boundary. Using available seismic data to (weakly) constrain the Moho surface, we invert the residual gravity field for topography of the base of the lithosphere. Lithosphere is found to be 200 km thick in the High Plains; 40–50 km in the eastern Great Basin; 75–100 km in the Colorado Plateau, and as thin as 40 km in the southern Rio Grande rift. In the area studied, the thickness of the lithosphere is everywhere greater than that of the crust. An asthenosphere bulge closely tracks the axis of the rift and is symmetrical to it, by contrast with asymmetry that might be inferred from broad features of the Bouguer anomaly. The separation of gravity effects made possible by the decompensative correction shows how the rift is fundamentally controlled by thinning of the lithosphere, yet in detail is deflected by long-lived tectonic welts in the shallow, brittle crust.

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...

INVESTIGATION OF MAGNETIZATION AND DENSITY OF A NORTH ATLANTIC SEAMOUNT USING POISSON’S THEOREM

The relationship between the gravitational and magnetic potentials caused by a uniform distribution of mass and magnetization may be used to obtain independent information about these physical properties. The general relationship in the frequency domain between the Fourier transforms of the gravity and magnetic anomaly fields is established through the Poisson theorem. The discrete Fourier transforms of the sampled continuous functions are used in an analysis which leads to a system of linear equations involving terms in density, magnetization, and calculated finite Fourier‐series coefficients. A least squares solution of the system yields the three components of the total magnetization vector divided by the density. From these results, the direction of total magnetization and the minimum of the Koenigsberger ratio Q can be determined uniquely. The remanent magnetization direction and certain other information can be derived for special cases in which the value of one or more of the physical property term...

Reconciliation of the discrete and integral Fourier transforms

In the application of harmonic analysis to potential‐field geophysical studies, relationships derived in terms of the continuous Fourier integral transform are evaluated in terms of the discrete Fourier transform. The discrete transform, obtained by transforming a finite number of equispaced samples of the actual aperiodic continuous function, is too low at the dc level and increasingly too high in the high frequencies, compared with the theoretical integral transform. As a consequence, overly restrictive limitations must be placed on high‐frequency‐amplifying operators such as differentiation and downward continuation. Also, a spurious and troublesome azimuthal distortion occurs in the discrete Fourier analysis of three‐dimensional (3-D) (map) data represented as grids. The discrete transform can be made essentially equivalent to the integral transform if, before sampling, the continuous aperiodic input function is made periodic by shifting the function by integer multiples of the data interval and summi...

Crustal extension in the Baikal rift zone

Analysis of the gravity field along four profiles crossing the Baikal rift zone permits an estimate of the amount of anomalous mass produced by 1. (1) graben-fill sediments, 2. (2) Moho uplift and intrusion of mantle sills and dikes, 3. (3) an asthenospheric bulge. Crustal extension is evaluated based on the idea of mass and volume balance of material introduced into and removed from the initial volume of the crust. Extension in the Baikal rift increases southwestward from 0.9 km in the Chara depression to 19.3 km in the South Baikal depression. These values generally agree with the position of the Euler pole determined from seismic data (fault plane solutions). Average rotation velocity for the lithospheric plates separated by the rift zone is estimated to be 5.93 × 10-4 rad/m.y. over about 30 m.y.