Inversion Of Potential Field Data Research Articles

Joint inversion of potential field data with adaptive unstructured tetrahedral mesh

Inverting potential field data presents a significant challenge due to its ill-posed nature, often leading to nonunique model solutions. Addressing this, our work focuses on developing a robust joint inversion method for potential field data, aiming to achieve more accurate density and magnetic susceptibility distributions. Unlike most previous work that used regular meshes, our approach adopts an adaptive, unstructured tetrahedral mesh, offering enhanced capabilities in handling the inverse problem of potential field methods. During inversion, the tetrahedral mesh is refined in response to the model update rate. We integrate a Gramian constraint into the objective function, allowing the enforcement of model similarity in terms of either the model parameters or their spatial gradients on an unstructured mesh. In addition, we use the moving least-squares method for gradient operator computation, which is essential for model regularization. Our model studies indicate that this method effectively inverts potential field data, yielding reliable subsurface density and magnetic susceptibility distributions. The joint inversion approach, compared with individual data set inversion, produces coherent geophysical models with enhanced correlations. Notably, it significantly mitigates the nonuniqueness problem, with the recovered anomaly locations aligning more closely with actual ground truths. Applying our methodology and algorithm to field data from the Ring of Fire area in Canada, the joint inversion process has generated comprehensive geophysical models with robust correlations, offering potential benefits for mineral exploration in the region.

Tomofast-x 2.0: an open-source parallel code for inversion of potential field data with topography using wavelet compression

Abstract. We present a major release of the Tomofast-x open-source gravity and magnetic inversion code that incorporates several functionalities enhancing its performance and applicability for both industrial and academic studies. The code has been re-designed with a focus on real-world mineral exploration scenarios, while offering flexibility for applications at regional scale or for crustal studies. This new version includes several major improvements: magnetisation vector inversion, inversion of multi-component magnetic data, wavelet compression, improved handling of topography with support for non-uniform grids, a new and efficient parallelisation scheme, a flexible parameter file, and optimised input–output operations. Extensive testing has been conducted on a large synthetic dataset and field data from a prospective area of the Eastern Goldfields (Western Australia) to explore new functionalities with a focus on inversion for magnetisation vectors and magnetic susceptibility, respectively. Results demonstrate the effectiveness of Tomofast-x 2.0 in real-world studies in terms of both the recovery of subsurface features and performances on shared and distributed memory machines. Overall, with its updated features, improved capabilities, and performances, the new version of Tomofast-x provides a free open-source, validated advanced and versatile tool for constrained gravity and magnetic inversion.

The Charnwood Terrane revisited: an integrated petrogenetic and petrophysical model for crustal structure in southern Britain

We based an integrated petrological and petrophysical model for the Ediacaran crust of southern Britain on a review of the c. 570–550 Ma Charnian volcano-sedimentary complex. The latter was emplaced in a magmatic rift wedge within the juvenile continental crust of the c. 720–600 Ma Marches Terrane, a subduction magmatic domain formed at the margin of the Gondwana palaeocontinent. The Charnian magmatic arc is characterized by primitive island arc tholeiite to more evolved calc-alkaline compositions. The inversion of aeromagnetic potential field data and petrophysical modelling reveals details of the internal structure of the Charnian Domain, including a median rift, superimposed annular structures and partitioning lineaments. The modelling suggests that the arc foundation could incorporate magnetite-rich cumulates, which may explain the anomalous geophysical properties, including crustal thickness, rigidity and buoyancy. There is no evidence for significant tectonic displacement between the Charnian Domain and its Marches Terrane host. Instead, the domain likely occupies a wedge-shaped arc/marginal rift basin complex, propagated from a neighbouring ocean into the Gondwana margin. Contemporaneous volcanic rift successions in the Welsh Borderland and Wales of the 570–550 Ma Charnian magmatic phase developed in coeval ensialic rifts within less strongly extended Marches Terrane lithosphere. A comparable diversity of subduction-related magmatism is found in the Neogene–recent Hikurangi destructive margin of New Zealand, providing a plausible analogue for Charnian magmatism. Supplementary material: Supplementary Publication 1 (borehole geophysical log correlation; petrophysical data table; unannotated geophysical maps and sections) and Supplementary Publication 2 (geochemistry analytical conditions and data table) are available at https://doi.org/10.6084/m9.figshare.c.6805248

HMCLab: a framework for solving diverse geophysical inverse problems using the Hamiltonian Monte Carlo method

SUMMARY The use of the probabilistic approach to solve inverse problems is becoming more popular in the geophysical community, thanks to its ability to address nonlinear forward problems and to provide uncertainty quantification. However, such strategy is often tailored to specific applications and therefore there is a need for common platforms to solve different geophysical inverse problems and showing potential and pitfalls of the methodology. In this work, we demonstrate a common framework within which it is possible to solve such inverse problems ranging from, for example, earthquake source location to potential field data inversion and seismic tomography. This allows us to fully address nonlinear problems and to derive useful information about the subsurface, including uncertainty estimation. This approach can, in fact, provide probabilities related to certain properties or structures of the subsurface, such as histograms of the value of some physical property, the expected volume of buried geological bodies or the probability of having boundaries defining different layers. Thanks to its ability to address high-dimensional problems, the Hamiltonian Monte Carlo (HMC) algorithm has emerged as the state-of-the-art tool for solving geophysical inverse problems within the probabilistic framework. HMC requires the computation of gradients, which can be obtained by adjoint methods. This unique combination of HMC and adjoint methods is what makes the solution of tomographic problems ultimately feasible. These results can be obtained with ‘HMCLab’, a numerical laboratory for solving a range of different geophysical inverse problems using sampling methods, focusing in particular on the HMC algorithm. HMCLab consists of a set of samplers (HMC and others) and a set of geophysical forward problems. For each problem its misfit function and gradient computation are provided and, in addition, a set of prior models can be combined to inject additional information into the inverse problem. This allows users to experiment with probabilistic inverse problems and also address real-world studies. We show how to solve a selected set of problems within this framework using variants of the HMC algorithm and analyse the results. HMCLab is provided as an open source package written both in Python and Julia, welcoming contributions from the community.

Spherical Planting Inversion of GRAIL Data

In large-scale potential field data inversion, constructing the kernel matrix is a time-consuming problem with large memory requirements. Therefore, a spherical planting inversion of Gravity Recovery and Interior Laboratory (GRAIL) data is proposed using the L1-norm in conjunction with tesseroids. Spherical planting inversion, however, is strongly dependent on the correct seeds’ density contrast, location, and number; otherwise, it can cause mutual intrusion of anomalous sources produced by different seeds. Hence, a weighting function was introduced to limit the influence area of the seeds for yielding robust solutions; moreover, it is challenging to set customized parameters for each seed, especially for the large number of seeds used or complex gravity anomalies data. Hence, we employed the “shape-of-anomaly” data-misfit function in conjunction with a new seed weighting function to improve the spherical planting inversion. The proposed seed weighting function is constructed based on the covariance matrix for given gravity data and can avoid manually setting customized parameters for each seed. The results of synthetic tests and field data show that spherical planting inversion requires less computer memory than traditional inversion. Furthermore, the proposed seed weighting function can effectively limit the seed influence area. The result of spherical planting inversion indicates that the crustal thickness of Mare Crisium is about 0 km because the Crisium impact may have removed all crust from parts of the basin.

Joint gravity and magnetic inversion with trans-dimensional alpha shapes and autoregressive noise models

Typical geophysical inverse problems are ill-posed and non-unique which causes challenges for interpretation. To address these issues, deterministic inversion methods often apply constraints to parameter values, which control the effective number of parameters. However, such approaches can inhibit inference on complex structural boundaries. Bayesian trans-dimensional (trans-D) parametrizations for Earth structure partition space based on data information with the ability to adapt the parametrization locally to data information. Therefore, trans-D approaches can avoid under- or over-parametrizing regions of the model. Nonetheless, these parametrizations depend on the choice of partitioning types, such as Voronoi nodes or wavelet decomposition. In addition, trade-offs exist between spatial resolution and correlated data errors. We present a hierarchical model that treats both spatial and data noise parametrizations as trans-D to better incorporate trade-offs between noise and structure into uncertainty quantification. This includes a hierarchical spatial partitioning based on linear and nearest-neighbor interpolations and alpha shapes. The alpha shapes provide advantages for the inversion of potential field data by permitting flexibility in the shapes of structures of interest. The trans-D autoregressive noise model quantifies the impact of correlated noise on geophysical parameter estimates. We compare these methods with nested Voronoi partitioning and show differences in uncertainties, data fit, and parsimony of the parametrizations. Studies on simulated data show well-resolved structures and successful decorrelation of data residuals while requiring few parameters. The inversion of field data infers basement and salt broadly consistent with previous studies, but results show additional details that are consistent with independent geological knowledge.

Structural, petrophysical, and geological constraints in potential field inversion using the Tomofast-x v1.0 open-source code

Abstract. The quantitative integration of geophysical measurements with data and information from other disciplines is becoming increasingly important in answering the challenges of undercover imaging and of the modelling of complex areas. We propose a review of the different techniques for the utilisation of structural, petrophysical, and geological information in single physics and joint inversion as implemented in the Tomofast-x open-source inversion platform. We detail the range of constraints that can be applied to the inversion of potential field data. The inversion examples we show illustrate a selection of scenarios using a realistic synthetic data set inspired by real-world geological measurements and petrophysical data from the Hamersley region (Western Australia). Using Tomofast-x's flexibility, we investigate inversions combining the utilisation of petrophysical, structural, and/or geological constraints while illustrating the utilisation of the L-curve principle to determine regularisation weights. Our results suggest that the utilisation of geological information to derive disjoint interval bound constraints is the most effective method to recover the true model. It is followed by model smoothness and smallness conditioned by geological uncertainty and cross-gradient minimisation.

Integration of magnetic, gravity, and well-logging data interpretation to delineate the structural framework and formation evaluation of Bahariya Formation, North Diyur area, northern Western Desert, Egypt

The Bahariya Formation in Egypt's Western Desert is a major source for minerals and hydrocarbon accumulation. It is also characterized by a relatively high radiation content because it contains iron oxide deposits that attract radioactive elements. The main objectives of our study are to establish depth to basement, basement configuration and related structural elements, and thickness and configuration of the overlain sedimentary section. In addition to the analysis of well-logging data, many advanced techniques have been applied to analyze magnetic and gravity data, including depth estimation, 2D magnetic and gravity modeling, and 3D inversion of potential field data. By integrating all available data, we can determine the structural control of the study area and evaluate the subsurface parameters. Well logging has been used for interpretation of porous and permeable zones, water saturation calculation, and basic lithology identification. The depth to basement in our study ranges from −1700 to −4500 m. The basement is shallow in the northern parts of the study area and deeper in the southern parts. The main clay minerals of the formation are montmorillonite, chlorite, and a mixed clay layer. The Bahariya Formation is composed mainly of sandy clay and sandstone, and therefore it is considered an excellent reservoir.

Improved Generalized Cross-Validation and Unbiased Predictive Risk Estimator Methods Using the RGSVD: Application to Inversion of Potential Field Data

The inversion of potential field data has widely utilized the generalized cross-validation (GCV) and the unbiased predictive risk estimator (UPRE) methods to determine the regularization parameter. However, these two methods are time-consuming and it is difficult for them to determine the optimal linear search range including the optimal regularization. To solve these problems, this article improves the GCV and UPRE methods using the RGSVD (randomized generalized singular value decomposition) algorithm. The improved methods first use the randomized algorithm to compute an approximate generalized singular value decomposition (GSVD) with less computational time. Then, the optimal linear search range is determined based on the generalized singular values. Finally, the GCV and the UPRE functions are efficiently computed on the basis of the results from the RGSVD algorithm. In this way, the GCV and UPRE methods using the RGSVD algorithm are able to determine the optimal regularization parameter fast and effectively. One comparative test shows the effectiveness and efficiency of the GCV and the UPRE methods using the RGSVD algorithm.

Gravity and magnetic joint inversion for basement and salt structures with the reversible-jump algorithm

SUMMARYGravity and magnetic data resolve the Earth with variable spatial resolution, and Earth structure exhibits both discontinuous and gradual features. Therefore, model parametrization complexity should be able to address such variability by locally adapting to the resolving power of the data. The reversible-jump Markov chain Monte Carlo (rjMcMC) algorithm provides variable spatial resolution that is consistent with data information. To address the prevalent non-uniqueness in joint inversion of potential field data, we use a novel spatial partitioning with nested Voronoi cells that is explored by rjMcMC sampling. The nested Voronoi parametrization partitions the subsurface in terms of rock types, such as sedimentary, salt and basement rocks. Therefore, meaningful prior information can be specified for each type which reduces non-uniqueness. We apply nonoverlapping prior distributions for density contrast and susceptibility between rock types. In addition, the choice of noise parametrization can lead to significant trade-offs with model resolution and complexity. We adopt an empirical estimation of full data covariance matrices that include theory and observational errors to account for spatially correlated noise. The method is applied to 2-D gravity and magnetic data to study salt and basement structures. We demonstrate that meaningful partitioning of the subsurface into sediment, salt, and basement structures is achieved by these advances without requiring regularization. Multiple simulated- and field-data examples are presented. Simulation results show clear delineation of salt and basement structures while resolving variable length scales. The field data show results that are consistent with observations made in the simulations. In particular, we resolve geologically plausible structures with varying length scales and clearly differentiate salt structure and basement topography.

A simple but efficient non-linear method for 2D inversion of magnetic field data based on Ridge-Regression algorithm

In geophysical exploration, inversion is carried out on the observed data to generate a geophysical model, approximating the subsurface geological struc­ture. In the interpretation of magnetic data, the subsurface model parameters are found by a proper inversion scheme. Hence, it will be possible to obtain the entire parameters of any features (e.g. Dike) including depth, width, and location. In this paper, theoretical and field studies were carried out to interpret the total components of magnetic anomalies of dikes at the finite depth. Moreover, a least-squares approach was used for depth determination using anomalous magnetic data. Potential field data inversion can be achieved through many optimization techniques. This study, however, it is attempted to develop an efficient two-dimensional (2D) inversion algorithm based on the Ridge Regression routine. The developed method was programmed using Matlab software and applied to three sets of synthetic magnetic data containing different percent of random noise to find out how good the results are. It was found that the proposed 2D inversion method can produce an accurate subsurface model that precisely explains the synthetic data in each case of data inversion. Finally, the method was applied to the real total magnetic field (TMF) data of Moghan Sedimentary basin. In that case, the estimated sedimentary basement depths were found to be in good agreement with that of the seismic data acquired before.

Crustal structure and basement-cover relationship in the Dangerous Grounds, offshore North-West Borneo, from 3D joint CSEM and MT imaging

Accurate mapping of crustal thickness variations and the boundary relationships between sedimentary cover rocks and the crystalline basement is very important for heat-flow prediction and petroleum system modeling of a basin. Using legacy industry 3D data sets, we investigated the potential of 3D joint inversion of marine controlled-source electromagnetic (CSEM) and magnetotelluric (MT) data incorporating resistivity anisotropy to map these parameters across subbasins in the Dangerous Grounds in the southwestern rifted margin of the South China Sea, where limited previous seismic and potential field basement interpretations are available for comparison. We have reconstructed 3D horizontal and vertical resistivity models from the seabed down to [Formula: see text] depth for a [Formula: see text] area. The resistivity-versus-depth profile extracted from our 3D joint inversion models satisfactorily matched the resistivity and lithologic well logs at a wildcat exploration well location chosen for model validation. We found that the maximum resistivity gradients in the computed first derivative of the 3D resistivity volumes predict a depth to basement that matches the acoustic basement. The models predict the presence of 2 to approximately 5 km thick electrically conductive ([Formula: see text]) sedimentary cover atop an electrically resistive ([Formula: see text]) crystalline crust that is underlain by an electrically conductive ([Formula: see text]) upper mantle at depths that vary laterally from approximately 25 to 30 km below sea level in our study area. Our resistivity variation with depth is found to be remarkably consistent with the density distribution at Moho depth from recent independent 3D gravity/gradiometry inversion studies in this region. We suggest that 3D joint inversion of CSEM-MT, seismic, and potential field data is the way forward for understanding the deep structure of such rifted margins.

A fast methodology for large-scale focusing inversion of gravity and magnetic data using the structured model matrix and the 2-D fast Fourier transform

SUMMARYWe 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.

Inverse modeling constrained by potential field data, petrophysics, and improved geologic mapping: A case study from prospective northwest Tasmania

The Heazlewood-Luina-Waratah area is a prospective region for minerals in northwest Tasmania, Australia, associated with historically important ore deposits related to the emplacement of granite intrusions and/or ultramafic complexes. The geology of the area is poorly understood due to the difficult terrain and dense vegetation. We have constructed an initial high-resolution 3D geologic model of this area using constraints from geologic maps and geologic and geophysical cross sections. This initial model is improved upon by integrating results from 3D geometry and physical property inversion of potential field (gravity and magnetic) data, petrophysical measurements, and updated field mapping. Geometry inversion reveals that the Devonian granites in the south are thicker than previously thought, possibly connecting to deep sources of mineralization. In addition, we identified gravity anomalies to the northeast that could be caused by near-surface granite cupolas. A newly discovered ultramafic complex linking the Heazlewood and Mount Stewart Ultramafic Complexes in the southwest also has been modeled. This implies a greater volume of ultramafic material in the Cambrian successions and points to a larger obducted component than previously thought. The newly inferred granite cupolas and ultramafic complexes are targets for future mineral exploration. Petrophysical property inversion reveals a high degree of variation in these properties within the ultramafic complexes indicating a variable degree of serpentinization. Sensitivity tests suggest maximum depths of 2–3 km for the contact aureole that surrounds major granitic intrusions in the southeast, whereas the Heazlewood River complex is likely to have a deeper source up to 4 km. We have demonstrated the value of adding geologic and petrophysical constraints to 3D modeling for the purpose of guiding mineral exploration. This is particularly important for the refinement of geologic structures in tectonically complex areas that have lithology units with contrasting magnetic and density characteristics.

OpenMP Implementation of a Novel Potential-Field-Data Source-Growth-Based Inversion Approach for 3D Salt Imaging in Deepwater Gulf of Mexico

Potential-field-data imaging of complex geological features in deepwater salt-tectonic regions in the Gulf of Mexico remains an open active research field. There is still a lack of resolution in seismic imaging methods below and in the surroundings of allochthonous salt bodies. In this work, we present a novel three-dimensional potential-field-data simultaneous inversion method for imaging of salt features. This new approach incorporates a growth algorithm for source estimation, which progressively recovers geological structures by exploring a constrained parameter space; restrictions are posed from a priori geological knowledge of the study area. The algorithm is tested with synthetic data corresponding to a real complex salt-tectonic geological setting commonly found in exploration areas of deepwater Gulf of Mexico. Due to the huge amount of data involved in three-dimensional inversion of potential field data, the use of parallel computing techniques becomes mandatory. In this sense, to alleviate computational burden, an easy to implement parallelization strategy for the inversion scheme through OpenMP directives is presented. The methodology was applied to invert and integrate gravity, magnetic and full tensor gradient data of the study area.

A comprehensive lithospheric study of Black Sea using thermal modeling and simultaneous joint 3D inversion of potential field data

We present a comprehensive lithospheric model of the Black Sea comprised of thermal structure, sediment thickness, and depths to Moho and lithosphere-asthenosphere boundary (LAB). The crustal structure (depths to Moho boundary and basement) is first estimated via a simultaneous joint 3D nonlinear inversion of satellite gravity data and the Gzz component of gravity gradiometry tensors. The results are then used for detailed 2D iterative forward modeling of potential field data based on the thermal structure of the lithosphere. In addition to potential field data, heat flow measurements are used for the iterative forward modeling. Eight 2D profiles across the Black Sea were selected to achieve a full coverage of the main geological structures. The average depth to basement is estimated as 10–11 km in the East Black Sea (EBS) basin and 11–12 km in the West Black Sea (WBS) basin. The average depth to basement beneath the ridges is estimated between 2 and 5 km. The modeling results show a mean thickness of 20 km and 21 km for the crust and 150 and 100 km for the lithosphere in the central parts of EBS and WBS, respectively. While the lithosphere is thicker in the EBS than in the WBS, a LAB uplift in the central basins is obvious in the reconstructed model. Also, the calculated surface heat flow is highest at the ridges while the lowest values are found over thick sediments in the central basins. After correcting the calculated heat flow data for the sedimentation effect, the resulting values match the published measurements indicating the reliability of the obtained thermal structure for the Black Sea.

Research Note: A unifying framework for the widely used stabilization of potential field inverse problems

ABSTRACTWe present a brief review of the widely used and well‐known stabilizers in the inversion of potential field data. These include stabilizers that use L2, L1 and L0 norms 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.

A short user guide for UBC-GIF gravity and magnetic inversions

Obtaining reliable predictions of the subsurface will provide a critical advantage for explorers seeking mineral deposits at depth and beneath cover. A common approach in achieving this goal is to use deterministic property-based inversion of potential field data to predict a 3D subsurface distribution of physical properties that explain measured gravity or magnetic data. The non-uniqueness of inversions of potential field data mandates careful and consistent parameterization of the problem to ensure realistic solutions. Including all prior geological knowledge as constraints on the inversion also helps ensure that the recovered predictions are consistent with both the geophysical data and the geological knowledge.

Regularized equidimension inversion of gravity data

The inversion of gravity data is to recover a 3-D distribution of density contrast based on the gravity observation data. In reality, we can merely utilize the limited and discrete gravity data known only on the surface in the inversion. That causes the inherent non-uniqueness and low depth resolution of the gravity data inversion results. In this paper, we developed a new method of the regularized equidimension inversion (REI), which achieved the inversion of the 3-D gravity anomaly model with 3-D gravity anomaly data. We combined the BG theory and Tikhonov regularization method, developed the regularized inversion of the potential field data. The character of equidimension inversion is to inverse the subsurface model utilizing the 3-D anomaly data. The 3-D anomaly is including the anomaly data of several height planes besides the 2-D anomaly data of the surface.

Reply to comment by S. Vatankhah and R.A. Renaut: “Three‐dimensional potential field data inversion with L0 quasinorm sparse constraints”

The comment on “Three-dimensional potential field data inversion with L0 quasi-norm sparse constraints” by Vatankhah mainly gives some advice on my paper. Vatankhah pays attention on the L0 norm, which is applied on the potential field data inversion. Last and Kubik (1983) firstly introduced the compact inversion of gravity data. Later, other inversion studies were based on this principle (e.g.: Portniaguine and Zhdanov 1999). The strategy to obtain compact models is similar to the L0 norm constraint. However, they are not same. In my paper, I used the L0 norm which was firstly introduced in the compressive sensing theory. Compressive sensing is the reconstruction of sparse images or signals from very few samples, by means of solving a tractable optimization problem (Baraniuk 2008; Chartrand 2009). It is a great breakthrough. Later, this principle has been applied in many domains (e.g., Mohimani, Babaie-Zadeh and Jutten 2009; Zhang and Tian 2017). In these researches, an approximate zero norm replaces the L0 norm. In Vatankhah’s comment, there may be some inaccurate understanding. Function f 2 σ (m) (equation (3) in Vatankhah’s comment, corresponding to the function used in Meng (2018)) is better suited for potential field data inversion comparing with f 1 σ (m) (equation (1) in Vatankhah’s comment, introduced by Last and Kubik, 1983). In Vatankhah’s Figure 1(a), for large values of σ and for big absolute values of m, f 2 σ (m) has smaller values than f 1 σ (m), and therefore f 2 σ (m) is smoother than f 1 σ (m). For large σ more weight is imposed on the large elements of the parameter vector m in f 1 σ (m), as compared with f 2 σ (m), yielding a larger penalty on these large elements of m. Thus, the f 2 σ (m) can provide smoother solutions. This phenomenon demonstrates that noise has less effect on f 2 σ (m) than on f 1 σ (m), both used to approximate the L0 norm. Therefore, the f 2 σ (m) function has better performance with big values of σ. Even when σ = 0.1 (Vatankhah’s Fig. 1(b)), f 2 σ (m) have smaller values than f 1 σ (m) for absolute values of m lesser than 0.2, and bigger values than f 1 σ (m) for absolute values ofmbetween 0.2 and 0.6. Therefore, equation (3) can give sparser and better inversion results. Vatankhah’s Figure 1 shows this phenomenon clearly. In the potential field data inversion, the depth-weighting function and density constraint function are very important and commonly used. I think that the researchers can focus on the solution on the L0 norm. For example, the Newton gradient method is applied in my paper, which can effectively avoid the calculations of step size. I think that this reply can help the readers to have a better understanding of this topic.