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.

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.

We report 26 new heat flow and 13 radiogenic heat production measurements from Zimbabwe, Zambia and Tanzania, together with details and some revisions of 18 previous heat flow measurements by other investigators from Kenya and Tanzania. These measurements come from Archean cratons, Proterozoic mobile belts, and Mesozoic and Cenozoic rifts. Heat flow data from eight new sites in the Archean Zimbabwe Craton are consistent with previous measurements in the Archean Kaapvaal‐Zimbabwe Craton and Limpopo Belt (Kalahari Craton) and do not change the mean heat flow of 47±2 mW m−2 (standard error of the mean) in the Kalahari Craton based on 53 previous measurements. Eight new sites in the Archean Tanzania Craton give a mean heat flow of 34±4 mW m−2. The mean heat flow from nine sites in the Proterozoic Mozambique Belt to the east of the Tanzania Craton in Kenya and Tanzania is 47±4 mW m−2. Twelve measurements in the Mesozoic rifted continental margin in east Africa give a mean heat flow of 68±4 mW m−2; four measurements in the Mesozoic Luangwa and Zambezi Rifts range from 44 to 110 mW m−2 with a mean of 76±14 mW m−2. In comparing heat flow in east and southern Africa, we observe a common heat flow pattern of increasing heat flow away from the centers of the Archean cratons. This pattern suggests a fundamental difference in lithospheric thermal structure between the Archean cratons and the Proterozoic and early Paleozoic mobile belts which surround them. Superimposed on this common pattern are two regional variations in heat flow. Heat flow in the Tanzania Craton is lower by about 13 mW m−2 than in the Kalahari Craton, and in the Mozambique Belt in east Africa heat flow is somewhat lower than in the southern African mobile belts at similar distances from the Archean cratonic margin. The two regional variations can be explained in several ways, none of which can as yet be elevated to a preferred status: (1) by variations in crustal heat production, (2) by thin‐skinned thrusting of the Mozambique Belt over the Tanzania Cratonic margin, (3) by lateral heat transfer from beneath the rift flanks into the rifts, or (4) by lower mantle heat flow beneath all of eastern Africa prior to the Cenozoic development of the East African rift system.

We present a new method for interpreting isolated potential‐field (gravity and magnetic) anomaly data. A linear equation, involving a symmetric anomalous field and its horizontal gradient, is derived to provide both the depth and nature of the buried sources. In many currently available methods, either higher order derivatives or postprocessing is necessary to extract both pieces of information; therefore, data must be of very high quality. In contrast, for gravity work with our method, only a first‐order horizontal derivative is needed and the traditional data quality is sufficient. Our proposed method is similar to the Euler technique; it uses a shape factor instead of a structural index to characterize the buried sources. The method is tested using theoretical anomaly data with and without random noise. In all cases, the method adequately estimates the location and the approximate shape of the source. The practical utility of the method is demonstrated using gravity and magnetic field examples from the United States and Zimbabwe.

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 gridbased version of Euler deconvolution that has the ability to define within an Euler operating window whether the source is 2D or 3D in character so that the solutions can be treated differently. We illustrate the new approaches on model and real data.