Alan Reid

Magnetic interpretation in three dimensions using Euler deconvolution

Magnetic‐survey data in grid form may be interpreted rapidly for source positions and depths by deconvolution using Euler’s homogeneity relation. The method employs gradients, either measured or calculated. Data need not be pole‐reduced, so that remanence is not an interfering factor. Geologic constraints are imposed by use of a structural index. Model studies show that the method can locate or outline confined sources, vertical pipes, dikes, and contacts with remarkable accuracy. A field example using data from an intensively studied area of onshore Britain shows that the method works well on real data from structurally complex areas and provides a series of depth‐labeled Euler trends which mark magnetic edges, notably faults, with good precision.

Euler deconvolution of gravity tensor gradient data

Tensor Euler deconvolution has been developed to help interpret gravity tensor gradient data in terms of 3-D subsurface geological structure. Two forms of Euler deconvolution have been used in this study: conventional Euler deconvolution using three gradients of the vertical component of the gravity vector and tensor Euler deconvolution using all tensor gradients. These methods have been tested on point, prism, and cylindrical mass models using line and gridded data forms. The methods were then applied to measured gravity tensor gradient data for the Eugene Island area of the Gulf of Mexico using gridded and ungridded data forms. The results from the model and measured data show significantly improved performance of the tensor Euler deconvolution method, which exploits all measured tensor gradients and hence provides additional constraints on the Euler solutions.

Magnetic source parameters of two‐dimensional structures using extended Euler deconvolution

The Euler homogeneity relation expresses how a homogeneous function transforms under scaling. When implemented, it helps to determine source location for particular potential field anomalies. In this paper, we introduce an additional relation that expresses the transformation of homogeneous functions under rotation. The combined implementation of the two equations, called here extended Euler deconvolution for 2-D structures, gives a more complete source parameter estimation that allows the determination of susceptibility contrast and dip in the cases of contact and thin-sheet sources. This allows for the structural index to be correctly chosen on the basis of a priori knowledge about susceptibility and dip. The pattern of spray solutions emanating from a single source anomaly can be attributed to interfering sources, which have their greatest effect on the flanks of the anomaly. These sprays follow different paths when using either conventional Euler deconvolution or extended Euler deconvolution. The paths of these spray solutions cross and cluster close to the true source location. This intersection of spray paths is used as a discriminant between poor and well-constrained solutions, allowing poor solutions to be eliminated. Extended Euler deconvolution has been tested successfully on 2-D model and real magnetic profile data over contacts and thin dikes.

New discrimination techniques for Euler deconvolution

Abstract Euler deconvolution has come into wide use as an aid to interpreting profile or gridded magnetic survey data. It provides automatic estimates of source location and depth. In doing this, it uses a structural index (SI) to characterise families of source types. Euler deconvolution can be usefully applied to gravity data. For simple bodies, the gravity SI is one less than the magnetic SI. For more complex bodies (including the contact case), the Euler method is at best an approximation. Extended Euler provides better-constrained solutions for both gravity and magnetic bodies. Seven alternative formulations on the extended Euler equations are presented. A parametric model study for extended Euler reveals the ability to calculate depths, SI, strike and error estimates. The computed SI values from model studies are compared to equivalent examples from field data across known geological structures. New discrimination techniques for isolating geological bodies of interest are proposed and applied. The computed SI is used to discriminate magnetic signatures arising from known kimberlites. In another study, the discrimination is used to classify major fault contacts. The advent of new formulations of Euler equations offers scope to further refine discrimination strategies.

Grid Euler deconvolution with constraints for 2D structures

The conventional formulation of 3D Euler deconvolution assumes that the observed field in each Euler window varies in all directions. Where the source is 2D, this assumption leads to the production of poorly constrained solutions. If the source is 2D, the problem leads to a rank deficient normal equations matrix having an eigenvector associated with a zero eigenvalue. This vector lies in the horizontal plane and is pointing along the strike direction, thus allowing for the identification of a 2D structure and its strike. Finding a pseudoinverse via eigenvector expansion allows accurate source location, and the strike information allows the automatic implementation of profile‐based techniques like extended Euler deconvolution to gridded data, thus allowing for the first time the estimation of strikes, dips, and susceptibilities from grids using an automatic process. We present a grid‐based version of Euler deconvolution that has the ability to define within an Euler operating window whether the source is 2...

Avoidable Euler Errors – the use and abuse of Euler deconvolution applied to potential fields†

Window-based Euler deconvolution is commonly applied to magnetic and sometimes to gravity interpretation problems. For the deconvolution to be geologically meaningful, care must be taken to choose parameters properly. The following proposed process design rules are based partly on mathematical analysis and partly on experience. 1. The interpretation problem must be expressible in terms of simple structures with integer Structural Index (SI) and appropriate to the expected geology and geophysical source. 2. The field must be sampled adequately, with no significant aliasing. 3. The grid interval must fit the data and the problem, neither meaninglessly overgridded nor so sparsely gridded as to misrepresent relevant detail. 4. The required gradient data (measured or calculated) must be valid,with sufficiently low noise, adequate representation of necessary wavelengths and no edge-related ringing. 5. The deconvolution window size must be at least twice the original data spacing (line spacing or observed grid spacing) and more than half the desired depth of investigation. 6. The ubiquitous sprays of spurious solutions must be reduced or eliminated by judicious use of clustering and reliability criteria, or else recognized and ignored during interpretation. 7. The process should be carried out using Cartesian coordinates if the software is a Cartesian implementation of the Euler deconvolution algorithm (most accessible implementations are Cartesian). If these rules are not adhered to, the process is likely to yield grossly misleading results. An example from southern Africa demonstrates the effects of poor parameter choices.

Euler deconvolution of gravity anomalies from thick contact/fault structures with extended negative structural index

The concept of extended Euler homogeneity of potential fields is examined with respect to all variables of length dimension in their analytical expressions.This reveals the possible existence of positive degrees of homogeneity or corresponding negative structural indices considered as extensions of theThompson’s structural indices in Euler deconvolution. This approach is implemented for a contact gravity model, represented by a 2D semi-infinite slab with large thickness relative to its depth.Applying Euler deconvolution on synthetic and field data indicates that the positive degree of homogeneity, i.e., the extended negative structural index, is the appropriate one for the inversion of gravity anomalies fromcontactstructures.

Euler Deconvolution of Gravity Data

Euler deconvolution of both profile and gridded magnetic data (Thompson, 1982; Reid et al, 1990) has found wide application. It has been implemented by many organisations and individuals and is commercially available from several suppliers. Applications to gravity are fewer. Marson & Klingele (1993) applied it to gravity vertical gradients. Keating applied it to gravity point data. Zhang et al (2000) applied it to gravity tensor gradient data. But Euler deconvolution requires a well-founded understanding of a critical parameter, the Structural Index (SI), which characterizes the source geometry. In the magnetic case it varies from zero (contact of infinite depth extent) to 3 (point dipole). It can be thought of as the index in the field strength fall-off with distance.

Euler Deconvolution: Past, Present, And Future-A Review

Hood (1963) showed that Euler’s relation could be used to calculate depth to point pole (SI=2) or point dipole (SI=3), given a measured vertical gradient. Ruddock et al. (1966) were awarded a U.S. patent describing the use of a vertical gradiometer and Euler’s relation to determine the depth of and fall-off rate (SI) from a magnetic discontinuity. They recognized SIs of 1, 2 and 3 as corresponding to sheets, line sources and point sources, respectively. The relation was subsequently employed to estimate source type, given position and depth known or estimated by other methods (Slack et al., 1967; Barongo, 1984). Steenland (1968) has pointed out that falloff rate (SI) is only approximately a constant for real source bodies over particular distance ranges. Thompson (1982) developed the profile technique quite fully, named it EULDPH, and suggested that SIs between 0.5 and 3 were useful on pole reduced magnetic data. The fault model (SI 0.5) required some empirically based corrections to obtain depth. Soon thereafter it was applied to data from the Witwatersrand Basin (Durrheim, 1983; Wilsher, 1987; Corner and Wilsher, 1989). Wilsher also showed, by application of Poisson’s relation, that the vertical gradient of gravity (i.e., pseudo-magnetic field) could be expected to behave like magnetic field and could benefit from EULDPH methods.

Euler magnetic structural index of a thin‐bed fault

Euler deconvolution (Thompson, 1982; Reid et al., 1990) has come into wide use as an aid to interpreting profile or gridded magnetic survey data. It provides automatic estimates of source location and depth. In doing this, it uses a structural index (SI) to characterize families of source types. Typical values are shown in Table 1.