I have developed an inversion approach that determines a 3D susceptibility distribution that produces a given magnetic anomaly. The subsurface model consists of a 3D, equally spaced array of dipoles. The inversion incorporates a model norm that enforces sparseness and depth weighting of the solution. Sparseness is imposed by using the Cauchy norm on model parameters. The inverse problem is posed in the data space, leading to a linear system of equations with dimensions based on the number of data, N . This contrasts with the standard least-squares solution, derived through operations within the M -dimensional model space ( M being the number of model parameters). Hence, the data-space method combined with a conjugate gradient algorithm leads to computational efficiency by dealing with an N×N system versus an M×M one, where N≪M . Tests on synthetic data show that sparse inversion produces a much more focused solution compared with a standard model-space, least-squares inversion. The inversion of aeromagnet...

3D magnetic data-space inversion with sparseness constraints

La Fossa cone is an active stratovolcano located on Vulcano Island in the Aeolian Archipelago (southern Italy). Its activity is characterized by explosive phreatic and phreatomagmatic eruptions producing wet and dry pyroclastic surges, pumice fall deposits, and highly viscous lava flows. Nine 2-D electrical resistivity tomograms (ERTs; electrode spacing 20 m, with a depth of investigation >200 m) were obtained to image the edifice. In addition, we also measured the self-potential, the CO2 flux from the soil, and the temperature along these profiles at the same locations. These data provide complementary information to interpret the ERT profiles. The ERT profiles allow us to identify the main structural boundaries (and their associated fluid circulations) defining the shallow architecture of the Fossa cone. The hydrothermal system is identified by very low values of the electrical resistivity ( 400 Ω m). Inside the crater it is possible to follow the plumbing system of the main fumarolic areas. On the flank of the edifice a thick layer of tuff is also marked by very low resistivity values (in the range 1–20 Ω m) because of its composition in clays and zeolites. The ashes and pyroclastic materials ejected during the nineteenth-century eruptions and partially covering the flank of the volcano correspond to relatively resistive materials (several hundreds to several thousands Ω m). We carried out laboratory measurements of the electrical resistivity and the streaming potential coupling coefficient of the main materials forming the volcanic edifice. A 2-D simulation of the groundwater flow is performed over the edifice using a commercial finite element code. Input parameters are the topography, the ERT cross section, and the value of the measured streaming current coupling coefficient. From this simulation we computed the self-potential field, and we found good agreement with the measured self-potential data by adjusting the boundary conditions for the flux of water. Inverse modeling shows that self-potential data can be used to determine the pattern of groundwater flow and potentially to assess water budget at the scale of the volcanic edifice.

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Inner structure of La Fossa di Vulcano (Vulcano Island, southern Tyrrhenian Sea, Italy) revealed by high-resolution electric resistivity tomography coupled with self-potential, temperature, and CO2 diffuse degassing measurements

[1] The self-potential (SP) method has long been used for a variety of geophysical applications because of its ease of acquisition, but has suffered from difficulty in interpretation of the data. Self-potential signals result from a source term that is coupled with the earth resistivity and appropriate boundary conditions. This work describes an inversion methodology for determining the self-potential sources from measured SP and resistivity data. The SP source inversion is a linear problem, though it is complicated by nonuniqueness that is common to potential-field problems. The linear operator is also poorly conditioned because of the limited set of measurements, which are often constrained to the earth's surface. Our approach utilizes model regularization that selects a class of solutions which fit the data with sources that are spatially compact. Large variations in sensitivity due to distance and resistivity structure throughout the model are addressed through the use of a scaling term derived from the Green's functions that define the linear operator. A significant benefit of these methods is the resolution of targets at depth from surface measurements alone. This inversion technique is first illustrated with a simple synthetic data set. In a second example we apply this approach to a field data set taken from previously published literature and investigate the effects of different resistivity structure assumptions on the inversion results. The spatial distribution of sources provides useful information that can subsequently be interpreted in terms of physical processes that generate the SP data.

https://agupubs.onlinelibrary.wiley.com/doi/pdf/10.1029/2006JB004262

Three-dimensional source inversion of self-potential data

3D gravity inversion using a model of parameter covariance

Nonparametric inverse methods provide a general framework for solving potentialfield problems. The use of weighted norms leads to a general regularization problem of Tikhonov form. We present an alternative procedure to estimate the source susceptibility distribution from potential field measurements exploiting inversion methods by means of a flexible depth-weighting function in the Tikhonov formulation. Our approach improves the formulation proposed by Li and Oldenburg (1996, 1998), differing significantly in the definition of the depth-weighting function. In our formalism the depth weighting function is associated not to the field decay of a single block (which can be representative of just a part of the source) but to the field decay of the whole source, thus implying that the data inversion is independent on the cell shape. So, in our procedure, the depth-weighting function is not given with a fixed exponent but with the structural index N of the source as the exponent. Differently than previous methods, our choice gives a substantial objectivity to the form of the depth-weighting function and to the consequent solutions. The allowed values for the exponent of the depth-weighting function depend on the range of N for sources: 0 ≤ N ≤ 3 (magnetic case). The analysis regarding the cases of simple sources such as dipoles, dipole lines, dykes or contacts, validate our hypothesis. The study of a complex synthetic case also proves that the depth-weighting decay cannot be necessarily assumed as equal to 3. Moreover it should not be kept constant for multi-source models but should instead depend on the structural indices of the different sources. In this way we are able to successfully invert the magnetic data of the Vulture area, Southern Italy. An original aspect of the proposed inversion scheme is that it brings an explicit link between two widely used types of interpretation methods, namely those assuming homogeneous fields, such as Euler deconvolution or depth from extreme points transformation and the inversion under the Tikhonovform including a depth-weighting function. The availability of further constraints, from drillings or known geology, will definitely improve the quality of the solution.

Inversion of potential field data using the structural index as weighting function rate decay

A three-dimensional (3D) inversion program is developed to interpret gravity data using a selection of constraints. This selection includes minimum distance, flatness, smoothness and compactness constraints, which can be combined using a Lagrangian formulation. A multigrid technique is also implemented to resolve separately large and short gravity wavelengths. The subsurface in the survey area is divided into rectangular prismatic blocks and the problem is solved by calculating the model parameters, i.e. the densities of each block. Weights are given to each block depending on depth, a priori information on density and the density range allowed for the region under investigation. The present computer code is tested on modelled data for a dipping dike and multiple bodies. Results combining different constraints and a weight depending on depth are shown for the dipping dike. The advantages and behaviour of each method are compared in the 3D reconstruction. Recovery of geometry (depth, size) and density distribution of the original model is dependent on the set of constraints used. From experimentation, the best combination of constraints for multiple bodies seems to be flatness and a minimum volume for the multiple bodies. The inversion method is tested on real gravity data from the Rouyn-Noranda (Quebec) mining camp. The 3D inversion model for the first 10 km is in agreement with the known major lithological contacts at the surface; it enables the determination of the geometry of plutons and intrusive rocks at depth.

Constraints in 3D gravity inversion

A three-dimensional (3D) electrical resistivity modelling code is developed to interpret surface and subsurface data Based on the integral equation, it calculates the charge density caused by conductivity gradients at each interface of the mesh, allowing the estimation of the potential everywhere without the need to interpolate between nodes Modelling generates a huge matrix, made up of Green's functions, which is stored by using the method of pyramidal compression The potential is compared with the analytical and the numerical solutions obtained by finite-difference codes for two models: the two-layer case and the vertical contact case The integral method is more accurate around the source point and at the limits of the domain for the potential calculation using a pole-pole array A technique is proposed to calculate the sensitivity (Jacobian) and Hessian matrices in 3D The sensitivity is based on the derivative with respect to the block conductivity of the potential computed using the integral equation; it is only necessary to compute the electrical field at the source location A direct extension of this technique allows the determination of the second derivatives The technique is compared with the analytical solutions and with the calculation of the sensitivity according to the method using the inner product of the current densities calculated at the source and receiver points Results are very accurate when the Green's function that includes the source image is used The calculation of the three components of the electric field on the interfaces of the mesh is carried out simultaneously and quickly, using matrix compression

Olivier Boulanger

Papers

Constraints in 3D gravity inversion

3D modelling and sensitivity in DC resistivity using charge density

3D regional scale modelling of the Flin Flon-Glennie Complex: preparatory data analysis and preliminary results, Manitoba and Saskatchewan

Application of remotely piloted aircraft systems (RPAS) to regional airborne geophysical mapping: a concept study