Gravity measurements have been used in a wide range of geophysical prospecting and investigations, such as in mineral explorations, engineering and environmental problems, and archeological site investigations (, p. 20). In general, gravity inversion is a process that is used to determine the density, size, shape and location of complex subsurface causative bodies from an observed gravity anomaly by using different mathematical modeling techniques. Thus, inversion of gravity data constitutes an important step in the quantitative interpretation, since the reconstruction of density contrast models markedly increases the amount of information that can be extracted from the gravity data.

However, a principal difficulty with the gravity data inversion is the inherent non-uniqueness and instability that also exist in any geophysical method (, p. 216). In other words, for the given observed gravity data, there are many equivalent density distributions that can reproduce the same field data. The standard approach used to select acceptable solutions that are geologically reasonable is to use additional information about the problem by making assumptions about the following aspects: (1) the model parameters (existing information on the subsurface structure from geological or other geophysical hindsight) and (2) the data parameters (statistical properties of the inexact data, e.g., Gaussian distribution of errors). Based on these assumptions, there are two approaches in gravity inversion. The first approach fixes the density and varies the geometry. This approach is nonlinear in nature and has been studied by many authors, for instance, , and . The second approach, which is the one used in this work, fixes the geometry and varies the density. This approach is linear in nature and has been investigated by many researchers .

In an effort to introduce more qualitative prior information, in particular developed a method called compact gravity inversion. Their strategy utilizes the compactness stabilizer to minimize the area (in 2D) or volume (in 3D) occupied by the causative body, which is equivalent to maximizing its compactness. generalized the compact inversion method by making use of compactness along several axes using Tikhonov's regularization. In 2006, Silva and Barbosa further developed the compact inversion method with the so-called “interactive inversion”, which estimates the location and geometry of several density anomalies. They simplified their old method to improve computational performance. The generalized compact and interactive inversion strongly need a priori information to yield an accurate estimation.

The compactness stabilizer , also known as the minimum-support stabilizer , has been borrowed and implemented by other researchers in various geophysical inversion methods . As was demonstrated by a number of researchers , this stabilizer is known to yield a compact or focused geophysical model with sharp boundaries. Apart from the inversion methods which produce focused images mentioned above, sparse geophysical inversion approaches derived from Lp-norm (0≤p≤1) stabilization have been developed by many researchers – for instance, the sparse seismic reflectivity inversion method , direct current resistivity data inversion algorithm , magnetic data sparse inversion method and sparse gravity data inversion technique , to mention only a few.

Some instability of the original compact gravity inversion algorithm of was reported by (1997, p. 87) when the data were contaminated with noise. Then (1997, p. 89) improved the original compact inversion by introducing a new approach to the 3D compact gravity inversion. The problem with the method of (1997, p. 89) arises when dealing with a multiple-source model, where the inversion algorithm tends to concentrate densities towards the surface regardless of the true depth of the causative bodies. In overcoming this drawback, improved the compact gravity inversion method by incorporating a new depth-weighting function. In this paper, we present a gravity inversion method that can produce compact and sharp images to assist the modeling of non-smooth, blocky geologic features with sharp boundaries. The proposed approach is based on the authors' previous work , to which the reader is referred for further details, with the following two main differences and advancements. The first is the proposal and incorporation of an auto-adaptive regularization and error-weighting function. This has improved the fast convergence of the method while keeping its stability. The second is the implementation of combined stopping criteria to terminate the iteration after an appropriate number of steps. The developed method uses an iteratively reweighted least-squares (IRLS) minimization algorithm in combination with an L0-norm stabilizer, depth-weighting and physical parameter inequality constraint to estimate a compact and sharp density contrast model of the subsurface.

To evaluate the functionality and efficiency of the method, the developed procedure was tested on several synthetic model examples. The examples presented here are randomly chosen to demonstrate the following: (I) the applicability of the proposed auto-adaptive regularization technique (Eq. 18) and error-weighting function (Eq. 16); (II) the performance of the method in producing compact and sharp images of the causative bodies; (III) the effectiveness of the combined stopping criterion. The forward and the inverse problem were carried out using the procedure described in the preceding sections. In the inversion of the synthetic examples, the same subsurface discretization as the one used in generating the synthetic data (forward modeling) is used. All the inversion tests are performed on a desktop computer (11th Gen Intel(R) Core(TM) i7-11700, 2.50 GHz processor). For the first and second synthetic examples presented in this work, (I) the model region was discretized into 60 × 15 rectangular cells, and the dimensions of each cell were taken as 10 × 10 m in the x and y directions respectively; (II) the synthetic gravity data were computed at 60 data points that are centered in each cell at the top side of the model to produce data at a 10 m sample interval; (III) the computed gravity data are contaminated with Gaussian noise that has a standard deviation that amounts to 4 % of the magnitude at each data point with zero mean .

The first synthetic data inversion has been done for the model presented in Fig. 3a. For this synthetic model, the causative bodies are two rectangular structures elongated differently in the horizontal and vertical directions and located at different depths. The causative bodies have the same density contrast of 1000 kg m-3. The density of the causative bodies are given relative to the zero density of uniform background. Figure 3a (upper panel) shows noise-free (solid line) and noise-contaminated (star dots) gravity data. Separate inversion runs for three different ℓo values (0.2, 0.3 and 0.4) were performed with the developed inversion method. Note that, for subsequent iterations, the proposed auto-adaptive regularization technique (Eq. 18) is used to compute ℓ for each case. At the beginning of the inversion, the iterations are initialized with ρF=0 and Wh=Wc=Wn=WL0=I. The lower-limit density contrasts of all cells is zero (ρmin=0), and the upper bound ρmax=1000 kg m-3.

The results of the inversion using the developed method for three different ℓo values are shown in Figs. 3b and 4. The corresponding data fit between the predicted (solid line) and the actual contaminated (stars) gravity data is also shown. Comparing the inversion results with the original synthetic model in Fig. 3a, the inversion has sufficiently recovered the true models. The depth, geometry and density distributions of the synthetic causative bodies were recovered adequately. This can confirm the applicability of the proposed auto-adaptive regularization technique (Eq. 18) and error-weighting function (Eq. 16). Notice that the results also indicate the robustness and stability of the developed inversion method for different ℓo values. The average computation time to finish the inversion is approximately 16.3 s.

The second synthetic model is more complicated and consists of two causative bodies placed at various depths. The bodies have different sizes, shapes and density contrasts. The first causative body is a vertical rectangular block, with density contrast 2000 kg m-3 and placed at 40 m depth, and the second body is a dipping dike, with density contrast 3000 kg m-3 at 20 m depth. The synthetic model is shown in the lower part of Fig. 5a, and the generated noise-corrupted and noise free-gravity data are shown on the upper part.

Using the generated synthetic data, the inversion was initiated by assigning an initial zero density to each cell. We set initial ℓo=0.3. The density contrast limits are bounded between the lower bound ρmin=0 and the upper bound ρmax=3000 kg m-3. Even though a maximum iteration of 20 was set, the misfit and smy between two consecutive iterations gradually fell below the threshold set after the 14th iteration. The total computation time was approximately 15.73 s.

In Fig. 5b, the resulting model from the inversion of the second synthetic model (Fig. 5a) using the proposed method is presented. As can be seen in Fig. 5b (upper panel), the modeled gravity data (solid line) fit adequately with the synthetic data. The result, presented in Fig. 5b (lower panel), indicates an acceptable reconstruction of the synthetic multi-sources and multi-shape bodies that are located at different depths. The true shape, location and density of the causative bodies are recovered adequately. Like the first example, the reproduced images of the localized multiple sources are compact and sharp (Fig. 5b, lower panel).

For the third and fourth synthetic examples, (I) the subsurface model was discretized into 100 × 20 rectangular cells; each cell has a size of 50 m in x and z directions; (II) the synthetic gravity data were computed on 100 data points with a sample spacing of 50 m. The third synthetic model includes two dipping dikes in opposite directions. The causative 2D bodies have different sizes and the same density contrast that amounts to 1000 kg m-3 in a homogeneous-background zero density. The top part of the shallower dipping dike lies at a depth of 200 m, and that of the deeper dike lies at a depth of 250 m. The computed gravity data were contaminated by uncorrelated Gaussian noise whose standard deviation was equal to 4 % of the difference between the maximum and the minimum anomaly and zero mean. The synthetic model and the corresponding data are shown in Fig. 6 at the lower and upper panels respectively.

The inversion process was commenced by setting the densities of all cells to zero. The initial value of ℓo was set to 0.4. The bounding density ranges were set to a minimum value ρmin=0 and maximum value ρmax=1000 kg m-3. The maximum number of iterations was set to 20. Here, the inversion converged after the 13th iteration, and the total computation time was approximately 66.49 s. The resulting model and the inverted data using the proposed method are shown in Fig. 7b. For the sake of comparison, keeping all inversion parameters the same, the synthetic data were also inverted with the classical L2-norm-regularized inversion approach, and the obtained result is shown in Fig. 7a. As can be seen from the lower panel of Fig. 7b, unlike the model in Fig. 7a, the developed method was able to produce a compact and sharp model successfully. The other concern, which can be seen from the result in Fig. 7a, is that the target density contrast values are underestimated in the case of the conventional L2-norm inversion. In contrast, the geometry, locations and densities of both anomalous structures were adequately recovered with the presented inversion method (see Fig. 7b).

The fourth synthetic model consists of two different rectangular, anomalous bodies (Fig. 8a, lower panel). The anomalous structures have different dimensions and are buried at different depths. The top of the first rectangular block is placed at a depth of 200 m, and its density contrast is -1000 kg m-3, while the top of the second block is placed at a depth of 250 m and has a density contrast of 1000 kg m-3. Different density contrast, size and depth of adjacent structures have been considered to show the ability of the presented inversion method in reconstructing true parameters for these models. In this synthetic example, the computed data are contaminated by Gaussian noise with a standard deviation of 3 % of the difference between the maximum and the minimum anomaly.

For the current example, the inversion process was initialized by setting the initial value of ℓo=0.5. The lower bound for the density constraint ρmin=-1000 kg m-3, and the upper bound ρmin=1000 kg m-3. Similar to the previous examples, though the maximum number of iterations was set to be 20, the iterative step terminated when the proposed combined criterion was satisfied after 11 iterations. The approximate running time required to finish the inversion was 55.64 s. Figure 8b lower panel shows the recovered density contrast model. The corresponding fits between synthetic (stars) and predicted data (line) are shown in the upper panel of the same figure. We can see that the recovered rectangular bodies are compact and have sharp boundaries. The obtained results also indicate that the depth and density contrast of the anomalous rectangular bodies have been determined sufficiently.

Here, the effectiveness and the advantage of the proposed combined stopping criterion are illustrated by comparing it with another commonly used stopping condition. For this reason, the inversion process was performed again with the developed inversion method using only the misfit function (|misfitk-1-misfitk|) as a stopping condition. Note that, for comparison purposes, all the other inversion parameters are set to be the same, except for the stopping criterion. The resulting recovered density contrast models and the data fit are presented in Fig. 9. The corresponding values of the misfit and smy as a function of iteration number are also shown in Fig. 10a. For the sake of comparison, the misfit and smy when using the proposed combined stopping criterion for the same data set are also presented in Fig. 10b. The stopping condition |misfitk-1-misfitk|≤μ was reached after 5 iterations, as shown in the curve of Fig. 10a, before the true density distribution had been recovered fully. In other words, the estimated models are not satisfactory because densities lower than the target density are observed around the edges of the anomalous bodies (Fig. 9). This indicates that, unlike the result presented in Fig. 8b, where the proposed combined stopping condition is used, quitting the iterative process only with |misfitk-1-misfitk|≤μ criterion produces a premature solution – that is, before the maximum compactness is achieved.

A number of other numerical experiments we carried out showed that there are situations where either misfitk| or |misfitk-1-misfitk| fall below the given threshold values, at earlier iterations, before the true density is fully recovered. Thus, it is hard to take only one criterion as a termination condition. As stated in Sect. 2.3.6, it has been mentioned that the same has also be pointed out in a number of previous works . On the other hand, in the case of the proposed criterion (that is, when both the conditions |smyk-1-smyk|≤τ and |misfitk-1-misfitk|≤μ are satisfied at the same time), the inversion process yields an acceptable model. This clearly illustrates the advantage of using the proposed stopping criterion and its effectiveness in quitting the iterative scheme after the optimal number of iterations.

To further illustrate the effectiveness of the proposed combined criterion, the inversion process is allowed to continue to the 16th iteration, and the model, as a result of this, is presented in Fig. 11a. The progressions of the misfit and smy in the course of the iterative procedure are also given in Fig. 11b. As can be seen from the result (Fig. 11b), the solution obtained at subsequent iterations after the 11th iteration, where the iteration is terminated with the proposed stopping condition, remains virtually unaltered. This can also be observed from the misfit and smy variation curves shown in Fig. 11b, such that after the 11th iteration the misfit and smy values remain literally unchanged. Moreover, the results also indicate the appropriateness of the suggested threshold values μ and τ used in the proposed stopping criterion. The other thing one can observe from the results in Fig. 11 is the stability of the developed inversion method. This can also illustrate the effectiveness of the newly proposed auto-adaptive regularization technique (Eq. 18) and error-weighting function (Eq. 16).

In general, the presented method was tested with noise-contaminated data that are generated from different geometries, locations, sizes and density contrasts of causative bodies, and it has successfully recovered all models. Moreover, all the reconstructed images of the presented synthetic models are compact and sharp. Numerous synthetic data inversions were performed to analyze the impact of the density contrast bounds. The obtained results, which are not presented here, suggest that the values of density contrast bounds have a significant effect on the results, and hence, to recover a feasible model, a good knowledge of the density bounds is vital. This has also been pointed out by number of authors, for example, , and in the case of inversion methods that use non-smooth stabilizers (L1-norm or L0-norm). Provided that the lower and upper density contrast bounds are chosen properly, this inversion technique produces acceptable solutions. Therefore, as was demonstrated using synthetic examples, the proposed method has effectively and efficiently recovered the synthetic models. Generally, the tests performed on different geometry synthetic models showed that the method gives acceptable results for localized multi-sources anomalies at different depths with sharp features.

To test the method in the real world, where the gravity data are contaminated with noise, the improved algorithm is implemented on gravity data acquired on different published geologic settings. The first one is taken from by carefully digitizing the residual gravity data. As it was given in , the data were measured over the Guichon Creek batholith in south-central British Columbia. For the details about the measurements and geology, the reader is referred to and . The residual gravity profile is digitized at regular intervals of 0.5 km to produce a total of 64 data points, as shown in Fig. 12 (star marks).

For the inversion, the source volume beneath the anomaly was divided into 64 × 22 square lattices, with the dimensions of each cell being 0.5 km in both the x and z directions. Based on the a priori information from , density values were constrained between the limits ρmin=-150 kg m-3 and ρmax=0.001 kg m-3. We start the inversion with a homogeneous initial model in which every block has the same zero density and an initial ℓo value of 0.48. The inversion was terminated after the ninth iteration because the stopping criteria were fulfilled. The resulting model is presented in Fig. 12b. For comparison, the results obtained by and by are also included in Fig. 12b. In addition, using the same inversion parameters, we have performed L2-norm-regularized inversion, and the obtained result is shown in Fig. 12a. The shape, real extent of the anomaly and depth to bottom from the developed method are very close to the true geological feature , which was obtained from drilling. That means the implementation of the presented method resulted in a better solution compared to and the conventional L2-norm inversion. Note that this reasonable result is obtained by using only the density contrast limits as a priori information.

The second test on measured gravity data is carried out using the published data by over the Woodlawn massive sulfide ore body, New South Wales, Australia. The residual anomaly of the area, consisting of 61 data measurements, sampled every 5 m, is digitized from . The details about the data measurement and the geology of the area are discussed in . The model subsurface was divided into 61 by 30 blocks with a dimension of 5 m in both the x and z directions. Inverse modeling was performed with bounding constraints ρmin=-600 and ρmax=1000 kg m-3. The initial given value for ℓo was 0.6. The final solution was obtained after the 11th iteration. The reconstructed model, including the final model of , is shown in Fig. 13. The cross-section of the ore body verified by drilling is also shown in the figure.

The recovered model is approximately coincident with the shape, depth of burial and density of the known ore body. Areas of misfits in the current and previous works are believed to be caused by the termination of the original data at both ends before having reached the background level. Thus, this can be additional evidence that the presented method can be successfully applied to real data.

We have presented an alternative gravity inversion method that can produce compact and sharp images by using the L0-norm-stabilizing functional that helps to model geological features with non-smooth, blocky geologic bodies. Physical parameter inequality constraints and depth weighting are integrated into the procedure. The method also incorporates an auto-adaptive regularization technique, which automatically determines a suitable regularization parameter at every iteration, and an error-weighting function that helps to improve both the stability and convergence of the method. One of the strongest sides of the proposed auto-adaptive regularization and error-weighting matrix is that they are not dependent on a priori knowledge of the noise level. Because of that, the method can yield reasonable results even when the noise level of the data is not known properly. We implemented a combined stopping criteria and illustrated its effectiveness to terminate the iterative inversion process after an optimal number of steps. To illustrate the efficiency and the capacity of the proposed procedure, numerous synthetic tests were done. From these, four synthetic examples were presented. According to the results from these synthetic examples, the method can be applied for multi-source localized bodies located at different depths and having different geometries with sharp features. Furthermore, the method proved to be efficient in resolving causative bodies both vertically and laterally and produced compact and sharp images. The obtained results also indicate that the method behaves well with different noise levels embedded in the data and still retains its stability. This can confirm the robustness and stability of the developed inversion method for different noise levels. The method was also tested on measured gravity data. We obtained geologically acceptable models, and the results showed that our approach is effective and reliable. From a computational point of view, the method is efficient and can be easily run on a personal computer in just a few seconds. In conclusion, the developed method is advantageous in that it is stable, efficient and resolves sharp subsurface futures with acceptable resolving capacity. In geophysical exploration, gravity data are more often used to image complex 3D structures of the subsurface; hence further development of the method to 3D is crucial. Accordingly, future work will deal with the extension of the presented method to a 3D gravity inversion algorithm.