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Abstract The Tell Atlas of Algeria has a huge potential for hydrothermal energy from over 240 thermal springs with temperatures up to$$98^\circ$$ ${98}^{\circ }$C in the Guelma area. The most exciting region is situated in the northeastern part which is known to have the hottest hydrothermal systems. In this work, we use a high-resolution gravity study to identify the location and origin of the hot water, and how it reaches the surface. Gravimetric data analysis shows the shapes of the anomalies arising due to structures at different subsurface depths. The calculation of the energy spectrum for the data also showcases the depths of the bodies causing anomalies. 3D-Euler deconvolution is applied to estimate the depths of preexisting tectonic structures (faults). These preprocessing steps assist with assessing signal attenuation that impacts the Bouguer anomaly map. The residual anomaly is used in a three-dimensional inversion to provide a subsurface density distribution model that illustrates the locations of the origin of the dominant subsurface thermal systems. Overall, the combination of these standard processing steps applied to the measurements of gravity data at the surface provides new insights about the sources of the hydrothermal systems in the Hammam Debagh and Hammam Ouled Ali regions. Faults that are key to the water infiltrating from depth to the surface are also identified. These represent the pathway of the hot water in the study area. 

SUMMARY A fast algorithm for the large-scale joint inversion of gravity and magnetic data is developed. The algorithm uses a non-linear Gramian constraint to impose correlation between the density and susceptibility of the reconstructed models. The global objective function is formulated in the space of the weighted parameters, but the Gramian constraint is implemented in the original space, and the non-linear constraint is imposed using two separate Lagrange parameters, one for each model domain. It is significant that this combined approach, using the two spaces provides more similarity between the reconstructed models. Moreover, it is shown theoretically that the gradient for the use of the unweighted space is not a scalar multiple of that used for the weighted space, and hence cannot be accounted for by adjusting the Lagrange parameters. It is assumed that the measured data are obtained on a uniform grid and that a consistent regular discretization of the volume domain is imposed. Then, the sensitivity matrices exhibit a block-Toeplitz-Toeplitz-block structure for each depth layer of the model domain, and both forward and transpose operations with the matrices can be implemented efficiently using two dimensional fast Fourier transforms. This makes it feasible to solve for large scale problems with respect to both computational costs and memory demands, and to solve the non-linear problem by applying iterative methods that rely only on matrix–vector multiplications. As such, the use of the regularized reweighted conjugate gradient algorithm, in conjunction with the structure of the sensitivity matrices, leads to a fast methodology for large-scale joint inversion of geophysical data sets. Numerical simulations demonstrate that it is possible to apply a non-linear joint inversion algorithm, with Lp-norm stabilisers, for the reconstruction of large model domains on a standard laptop computer. It is demonstrated, that while the p = 1 choice provides sparse reconstructed solutions with sharp boundaries, it is also possible to use p = 2 in order to provide smooth and blurred models. The methodology is used for inverting gravity and magnetic data obtained over an area in northwest of Mesoproterozoic St Francois Terrane, southeast of Missouri, USA. 

ABSTRACT We present a brief review of the widely used and well‐known stabilizers in the inversion of potential field data. These include stabilizers that useL₂,L₁andL₀norms of the model parameters and the gradients of the model parameters. These stabilizers may all be realized in a common setting using two general forms with different weighting functions. Moreover, we show that this unifying framework encompasses the use of additional stabilizations which are not common for potential field inversion. 

l1 regularization is used to preserve edges or enforce sparsity in a solution to an inverse problem. We investigate the split Bregman and the majorization-minimization iterative methods that turn this nonsmooth minimization problem into a sequence of steps that include solving an -regularized minimization problem. We consider selecting the regularization parameter in the inner generalized Tikhonov regularization problems that occur at each iteration in these iterative methods. The generalized cross validation method and chi2 degrees of freedom test are extended to these inner problems. In particular, for the chi2 test this includes extending the result for problems in which the regularization operator has more rows than columns and showing how to use the -weighted generalized inverse to estimate prior information at each inner iteration. Numerical experiments for image deblurring problems demonstrate that it is more effective to select the regularization parameter automatically within the iterative schemes than to keep it fixed for all iterations. Moreover, an appropriate regularization parameter can be estimated in the early iterations and fixed to convergence. 

An efficient algorithm for the Lp -norm joint inversion of gravity and magnetic data using the cross-gradient constraint is presented. The presented framework incorporates stabilizers that use Lp -norms ( 0≤p≤2 ) of the model parameters, and/or the gradient of the model parameters. The formulation is developed from standard approaches for independent inversion of single data sets, and, thus, also facilitates the inclusion of necessary model and data weighting matrices, for example, depth weighting and hard constraint matrices. Using the block Toeplitz Toeplitz block structure of the underlying sensitivity matrices for gravity and magnetic models, when data are obtained on a uniform grid, the blocks for each layer of the depth are embedded in block circulant circulant block matrices. Then, all operations with these matrices are implemented efficiently using 2-D fast Fourier transforms, with a significant reduction in storage requirements. The nonlinear global objective function is minimized iteratively by imposing stationarity on the linear equation that results from applying linearization of the objective function about a starting model. To numerically solve the resulting linear system, at each iteration, the conjugate gradient algorithm is used. This is improved for large scale problems by the introduction of an algorithm in which updates for the magnetic and gravity parameter models are alternated at each iteration, further reducing total computational cost and storage requirements. Numerical results using a complicated 3-D synthetic model and real data sets obtained over the Galinge iron-ore deposit in the Qinghai province, north-west (NW) of China, demonstrate the efficiency of the presented algorithm. 

SUMMARY We discuss the focusing inversion of potential field data for the recovery of sparse subsurface structures from surface measurement data on a uniform grid. For the uniform grid, the model sensitivity matrices have a block Toeplitz Toeplitz block structure for each block of columns related to a fixed depth layer of the subsurface. Then, all forward operations with the sensitivity matrix, or its transpose, are performed using the 2-D fast Fourier transform. Simulations are provided to show that the implementation of the focusing inversion algorithm using the fast Fourier transform is efficient, and that the algorithm can be realized on standard desktop computers with sufficient memory for storage of volumes up to size n ≈ 106. The linear systems of equations arising in the focusing inversion algorithm are solved using either Golub–Kahan bidiagonalization or randomized singular value decomposition algorithms. These two algorithms are contrasted for their efficiency when used to solve large-scale problems with respect to the sizes of the projected subspaces adopted for the solutions of the linear systems. The results confirm earlier studies that the randomized algorithms are to be preferred for the inversion of gravity data, and for data sets of size m it is sufficient to use projected spaces of size approximately m/8. For the inversion of magnetic data sets, we show that it is more efficient to use the Golub–Kahan bidiagonalization, and that it is again sufficient to use projected spaces of size approximately m/8. Simulations support the presented conclusions and are verified for the inversion of a magnetic data set obtained over the Wuskwatim Lake region in Manitoba, Canada.