In this paper the 3D inversion of gravity data using two different regularization methods, namely Tikhonov regularization and truncated singular value decomposition (TSVD), is considered. The earth under the survey area is modeled using a large number of rectangular prisms, in which the size of the prisms are kept fixed during the inversion and the values of densities of the prisms are the model parameters to be determined. A depth weighting matrix is used to counteract the natural decay of the kernel, so the inversion obtains reliable information about the source distribution with respect to depth. To generate a sharp and focused model, the minimum support (MS) constraint is used, which minimizes the total area with non zero departure of the model parameters from a given a priori model. Then, the application of iteratively reweighted least square algorithm is required to deal with non-linearity introduced by MS constraint. At each iteration of the inversion, a priori variable weighting matrix is updated using model parameters obtained at the previous iteration. We use the singular value decomposition (SVD) for computing Tikhonov solution, which also helps us to compare the results with the solution obtained by TSVD. Thus, the algorithms presented here are suitable for small to moderate size problems, where it is feasible to compute the SVD. In Tikhonov regularization method, the optimal regularization parameter at each iteration is obtained by application of the x2 - principle parameter-choice method. The method is based on the statistical distribution of the minimum of the Tikhonov function. For weighting of the data fidelity by a known Gaussian noise distribution on the measured data and, when the regularization term is considered to be weighted by unknown inverse covariance information on the model parameters, the minimum of the Tikhonov functional becomes a random variable that follows a x2 distribution. Then, a Newton rootfinding algorithm can be used to find the regularization parameter. For truncated SVD regularization, the Picard plot is used to find a suitable value of truncation index. In math literature, a plot of singular values together with SVD and solution coefficients is often referred to as Picard plot. To test the algorithms, a density model which consists of a dipping dike embedded in a uniform half-space is used. The surface gravity anomaly produced by this model is contaminated with three different noise levels, and are used as input for introduced inversion algorithms. The results indicate that the algorithms are able to recover the geometry and density distribution of the original model. In general, the reconstructed model is more sparse using TSVD method as compare with Tikhonov solution. This especially happens for high noise level, where there is an important difference between two solutions. In this case, while TSVD produces a sparse model, the solution of Tikhonov regularization is not sparse. Furthermore, the number of iterations, which is required to terminate the algorithms, is more for TSVD as compare with Tikhonov method. This feature, along with automatic determination of regularization parameter, makes the implementation of the Tikhonov regularization method faster than TSVD. The inversion methods are used on real gravity data acquired over the Gotvand dam site in the south-west of Iran. Tertiary deposits of the Gachsaran formation are the dominant geological structure in this area, and it is mainly comprised of marl, gypsum, anhydrite and halite.There are several solution cavities in the area so that relative negative anomalies are distinguishable in the residual map. A window of residual map consists of 640 gridded data, which includes three negative anomalies, that is selected for modeling. The reconstructed models are shown and compare with results obtained by bore holes.

In this paper, we introduce a new linear received signal strength-based estimator for unknown node localization which its accuracy at low Signal-to-noise ratio (SNR) is better than many linear estimators and can compete with estimators based on the convex optimization, but it is much lighter than convex optimization-based estimators. The main ingredients in our proposed linear position estimator are to reformulate the localization problem in terms of Tikhonov-regularization and introduce a biased noise variable. The way that we apply for this reformulation avoids any possible linear approximation in which target position variables are involved, thus saving fair amount of information. The proposed algorithm is also indifferent to the transmit power and thus, applicable to either known or unknown transmit power scenarios. Simulation results show the efficacy of the proposed algorithm in comparison to the other methods for both typical RSS-based measurement data model and the modified model for indoor application.

summary Downward continuation of potential field data plays an important role in interpretation of gravity and magnetic data. For its inherent instability, many methods have been presented to downward continue stably and precisely. The Tikhonov regularization approach is one of the most robust. It is based on a lowpass filter derivation in the Fourier spectral domain, by means of a minimization problem solution. In this manuscript, we propose an improved regularization operator for downward continuation of potential field data. First, we simply define a special wavenumber named the cutoff wavenumber to divide the potential field spectrum into the signal part and the noise part based on the radially averaged power spectrum of potential field data. Next, we use the conventional downward continuation operator to downward continue the signal and the Tikhonov regularization operator to suppress the noise. Moreover, the parameters of the improved operator are defined by the cutoff wavenumber which has an obvious physical significance. For computing the α parameter, it is necessary that the C-norm of the potential field must be calculated. The improved operator can not only eliminate the influence of the high-wavenumber noise but also avoid the attenuation of the signal. Experiments through synthetic gravity and real gravity data from Kohe Namak region, Ghom province, Iran show that the downward continuation precision of the proposed operator is higher than the Tikhonov regularization operator...

The methods applied to regularization of the ill-posed problems can be classified under “direct” and “indirect” methods. Practice has shown that the effects of different regularization techniques on an ill-posed problem are not the same, and as such each ill-posed problem requires its own investigation in order to identify its most suitable regularization method. In the geoid computations without applying Stokes formula, the downward continuation based on Abel-Poisson integral is an inverse problem, which requires regularization. Since so far the regularization of this ill-posed problem has been thoroughly studied, in this paper the regularization of the downward continuation problem based on Abel-Poisson integral, is investigated and various techniques falling into the aforementioned classes of regularizations are applied and their efficiency is compared. From the first class Truncated Singular Value Decomposition (TSVD) and Truncated Generalized Singular Value Decomposition (TGSVD) methods and from the second class Generalized Tikhonov (GT) with the norms and semi-norms in Sobolev subspaces W12(a,b), W22(a,b) are applied and their capabilities for the regularization of the problem is compared. Our numerical results derived from simulated studies reveal that the GT method with discretized norm of Sobolev subspace W22(a,b) gives the best results among the studied methods for the regularization of the downward continuation problem based on the Abel-Poisson integral. On the contrary, the TGSVD method with the discretized second order derivatives has less consistency with the ill-posed problem and yields less accuracy. Finally, the GT method with discretized norm of Sobolev subspace W22(a,b) is applied to the downward continuation of real gravity data of the type modulus of gravity acceleration within the geographical region of Iran to derive a geoid model for this region.

Summary Two-dimensional (2-D) surface wave tomography is one of the most practical approach to reveal the lateral variation of phase/group velocity. Hence, various methods have been introduced to attain the more accurate images. In this paper, we compare two different tomography methods (Tikhonov regularization method and Yanovskaya-Ditmar method) and try to show their advantages and drawbacks resulted from their basic assumptions. We first investigate the lateral resolution capability of the methods using the synthetic checkerboard test in cases of noise free and noise level of 3%. Synthetic models are considered with different grid spacings of 0. 5˚ ×0. 5˚ , 1˚ ×1˚ and 2˚ ×2˚ . Then, they were applied on real data containing phase velocity obtained from the teleseismic Rayleigh waves recorded at broad band stations located in Iran at the period of 30 s. A comparison of the results obtained from the Tikhonov regularization and Yanoskaya-Ditmar methods shows that the former method has a better lateral resolution than the latter method...

Gravity inversion methods play a fundamental role in subsurface exploration, facilitating the characterization of geological structures and economic deposits. In this study, we conduct a comparative analysis of two widely used regularization methods, Tikhonov (L2) and Sparse (L1) regularization, within the framework of gravity inversion. To assess their performance, we constructed two distinct synthetic models by implementing tensor meshes, considering station spacing to discretize the subsurface environment precisely. Both methods have proven ability to recover density distributions while minimizing the inherent non-uniqueness and ill-posed nature of gravity inversion problems. Tikhonov regularization yields stable results, presenting smooth model parameters even with limited prior information and noisy data. Conversely, sparse regularization, utilizing sparsity-promoting penalties, excels in capturing sharp geological features and identifying anomalous regions, such as mineralized zones. Applying these methodologies to real gravity data from the Safu manganese deposit in northwest Iran, we assess their efficacy in recovering the geometry of dense ore deposits. Sparse regularization demonstrates superior performance, yielding lower misfit values and sharper boundaries during individual inversions. This underscores its capacity to provide a more accurate representation of the depth and edges of anomalous targets in this specific case. However, both methods represent the same top depth of the target in the real case study, but the lower depth and density distribution were not the same in the XZ cross-sections. Inversion results imply the presence of a near-surface deposit characterized by a high-density contrast and linear distribution, attributed to the high grade of manganese mineralization.