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2.5D regularized inversion for the interpretation of residual gravity data by a dipping thin sheet: numerical examples and case studies with an insight on sensitivity and nonuniqueness
 Salah A. Mehanee^{1}Email author and
 Khalid S. Essa^{1}
 Received: 26 March 2015
 Accepted: 27 June 2015
 Published: 18 August 2015
Abstract
A new twoandahalf dimensional (2.5D) regularized inversion scheme has been developed for the interpretation of residual gravity data by a dipping thinsheet model. This scheme solves for the characteristic inverse parameters (depth to top z, dip angle θ, extension in depth L, strike length 2 Y, and amplitude coefficient A) of a model in the space of logarithms of these parameters (log(z), log(θ), log(L), log(Y), and log(A)). The developed method has been successfully verified on synthetic examples without noise. The method is found stable and can estimate the inverse parameters of the buried target with acceptable accuracy when applied to data contaminated with various noise levels. However, some of the inverse parameters encountered some inaccuracy when the method was applied to synthetic data distorted by significant neighboring gravity effects/interferences. The validity of this method for practical applications has been successfully illustrated on two field examples with diverse geologic settings from mineral exploration. The estimated inverse parameters of the real data investigated are found to generally conform well with those yielded from drilling. The method is shown to be highly applicable for mineral prospecting and reconnaissance studies. It is capable of extracting the various characteristic inverse parameters that are of geologic and economic significance, and is of particular value in cases where the residual gravity data set is due to an isolated thinsheet type buried target. The sensitivity analysis carried out on the Jacobian matrices of the field examples investigated here has shown that the parameter that can be determined with the superior accuracy is θ (as confirmed from drilling information). The parameters z, L, Y, and A can be estimated with acceptable accuracy, especially the parameters z and A. This inverse problem is nonunique. The nonuniqueness analysis and the tabulated inverse results presented here have shown that the parameters most affected by the nonuniqueness are L and Y. It has also been shown that the new scheme developed here is advantageous in terms of computational efficiency, stability and convergence than the existing gravity data inversion schemes that solve for the characteristic inverse parameters of a sheet/dike.
Keywords
 Regularized 2.5D residual gravity data inversion
 3D thinsheet inversion
 Logspace inversion
 Nonuniqueness analysis
 Convergence analysis
 Sensitivity analysis
Background
Gravity and magnetic methods have many successful applications in mineral prospecting, environmental applications, and crustal imaging (e.g., Abdelrahman et al. 1989; Ateya and Takemoto 2002; BatistaRodríguez et al. 2013; Beiki and Pedersen 2011; Fedi 2007; Hinze 1990; Hinze et al. 2013; LaFehr and Nabighian 2012; Long and Kaufmann 2013; Mehanee 2014a; Nettleton 1976; Okubo et al. 2013; Paoletti et al. 2013; Pei et al. 2014).
Techniques based on some geometrically simple bodies (e.g., sheet) have been in use for the analysis of magnetic and gravity data acquired along profiles since the 1960s (e.g., Grant and West 1965). These techniques (e.g., Ateya and Takemoto 2002; Mushayandebvu et al. 2001; Sazhina and Grushinsky 1971) are applied to obtain the characteristic inverse parameters of the underlying target.
Inverse interpretation of an isolated residual gravity anomaly by a 3D thinsheet model remains of interest in geophysical prospecting (e.g., Grant and West 1965; Telford et al. 1976). The main objective of the interpretation in this case is to retrieve the characteristic inverse parameters of the 3D approximative model: depth to the top, thickness, extension in depth, extension in the strike direction, and direction and amount of dip.
Grant and West (1965) described a graphical method for the interpretation of a residual gravity anomaly measured along a profile by a 3D dipping thinsheet model. The approach was successfully applied to sulfide prospecting. However, the drawback with this method is that it is based on sets of main and auxiliary characteristic curves, and it essentially demands interpolation between pairs of characteristic curves in order to estimate the magnitudes of the sheet parameters. Telford and Geldart (1976) presented and described the forward modeling formula of 3D thinsheet model, which is the basis for the inversion scheme developed here. They also highlighted the limitations (which are reported above) of this formula. Holstein et al. (2010) used the concept of gravimagnetic similarity, and extended the thinsheet potential modeling formula to include the potential, field, and field gradient in both gravity and magnetic cases when the buried bodies have uniform density or magnetization. Holstein and Anastasiades (2010) derived an exact finite expansion for the forward modeling computation of the gravitational anomaly of a uniform thinpolygonal sheet. This expansion exhibits the required absolute numerical stability (Holstein and Anastasiades 2010). One can refer to Holstein and Ketteridge (1996) and Holstein et al. (2014), and the references therein, for a more detailed and comprehensive information on this particular thinpolygonal sheet subject. Beiki and Pedersen (2011) developed a data window constrained twodimensional (2D) inversion technique for the interpretation of gravity gradient tensor data using dike/sheet and contact models. By definition, a data window contains a number of particular data points out of the entire measured gravity profile (Beiki and Pedersen 2011). These particular data points are measured with respect to the center (that is located at the maximum response of the profile) of this window. This means that the gravity data points located to the left and right flanks of a window were not considered in the inversion. In other words, the full measured gravity signature (that is, primarily influenced by the buried anomalous structure) was not considered in the interpretation. A series of successive data windows is constructed by increasing the number of data points used (out of the whole gravity profile) for estimating the target parameters (Beiki and Pedersen 2011). Beiki and Pedersen (2011) used the MATLAB nonlinear least squares optimization tool “lsqnonlin” to solve the minimization problem of their scheme. This minimization tool is based on the LevenbergMarquardt method (Levenberg 1944; Marquardt 1963). The convergence of the LevenbergMarquardt method and its corresponding inverse parameters (e.g., the depth, dip angle, and thickness of the buried target) obtained from each window were plotted in their paper. According to Beiki and Pedersen (2011), solution with the smallest data fit error is selected (out of the entire solutions retrieved from all attempted data windows) as the most reliable one. It appears from the numerical models analyzed by Beiki and Pedersen (2011) that the use of a few data points (out of the whole measured gravity profile) in inversion was essential so that the LevenbergMarquardt method can converge. Note that the aforementioned limitations (that are pertinent to the number of data points used in inversion and to the convergence and stability of the minimizer) are satisfactorily treated and fully well illustrated here as will be seen. Note that the LevenbergMarquardt method (Madsen et al. 2004) was used also in the seismic tomography inversions.
Holstein et al. (2014) derived the gravity potential, field, and field gradient for triangular sheet targets with continuous densities across the boundaries to make it possible to approximate general curved surfaces. They verified the obtained modeling formula via numerical testing and simulated a model described by a dike curvature and a density compaction.
From this review, it appears to us that Grant and West (1965) are the only authors to have developed a method for the quantitative inverse interpretation of a residual gravity profile measured over a 3D dipping structure by 3D dipping thinsheet model. Consequently, to the best of our knowledge, a regularized scheme (based on 3D dipping thinsheet type model) for the inversion of residual gravity anomaly measured along a profile over a 3D body was not developed before.
The objective of this paper is to develop an efficient and rigorous regularized inversion scheme (based on 3D dipping thinsheet model) for the interpretation of a residual gravity anomaly measured along a profile traversing the center of a 3D buried anomalous structure and normal to the strike of this structure. This inversion scheme is called twoandahalf dimensional (2.5D) inversion (see e.g., Hinze et al. 2013, p. 177). The developed scheme is capable of dealing with the nonlinearity and the illposedness of this 2.5D inverse problem and has many merits. First, it inverts the entire residual gravity data set rather than just a few characteristic points out of this data set. Second, it simultaneously recovers all the characteristic inverse parameters (depth z, direction and amount of dip θ, extension in depth L, strike length 2 Y, and amplitude coefficient A) of the model (Fig. 1). Note that the parameter A is given by Δ ρ t (A=Δ ρ t), where Δ ρ and t are the density contrast and thickness of the sheet (a discussion on this parameter is given in the “Forward modeling solution” section), respectively. Third, the algorithm uses the exact forward modeling formula and evaluates the Jacobian (Fréchet) matrix (sensitivity matrix) of the scheme analytically and accurately. Fourth, the developed scheme (referred here as the logarithmicspace algorithm) employs the Tikhonov regularization (Tikhonov and Arsenin 1977) and the steepest descent (SD) and GaussNewton (GN) methods (e.g., Zhdanov 2002) in the space of logarithms of the model parameters (log(z), log(θ), log(L), log(Y), and log(A)) rather than in the space of the parameters themselves (z,θ,L,Y, and A). The use of the logarithmed parameters is essential in order to impose the positivity property of these parameters and hence to maintain the convergence and stability of the inversion scheme. The main achievements of this paper are as follows: (1) the use of the logarithm in all inverse parameters of the anomalous body in order to maintain the positivity of these parameters and the convergence of the inversion scheme, (2) the investigation of the nonuniqueness (that turned out to be imperative as will be seen and discussed here) of this particular inverse problem, and (3) the insights pertinent to the model parameters sensitivities in relation to non uniqueness.
The paper is structured as follows. First, we briefly describe the direct problem (“Forward modeling solution” subsection). Second, the formulation of this particular inverse problem and its solving is discussed. Third, before applying this method to real data, the accuracy of the scheme is assessed and verified to numerical models with and without noise. Finally, the applicability of the developed technique to real data is demonstrated on two published field data examples from mineral exploration.
Methods
Forward modeling solution
where x _{ i } is the coordinate (m) of the measurement station, Δ ρ is the density contrast (kg.m^{3}), t is the thickness (m), θ is the dip angle (degrees) of the body measured anticlockwise, z is the depth (m) to the top, 2 Y is the strike length (m), L is the dipping extent (m) of the body, and γ is the gravitational constant (6.67384 × 10^{11} m ^{3} kg ^{−1} s^{2}).
As indicated in the “Background” section, the thin sheet described above is a good approximation to a 3D prismatic structure, unless the thickness is somewhat greater than the depth to the top (z) (Telford and Geldart 1976). The presence of the thickness (t) outside the main square bracket of formula (1) is a consequence of the thinsheet assumption used in deriving this formula.
Note that formula (1) can be used to accurately simulate a twodimensional (2D) target by substituting in this formula a large value for the parameter Y (Grant and West 1965). Analysis of formula (1) with respect to the 2D simulation accuracy is beyond the scope of this paper.
It is noted that the parameters Δ ρ and t can be combined into a single parameter (the socalled amplitude coefficient, A (kg.m^{2}) =Δ ρ t) to help minimize the nonuniqueness nature of this particular inverse problem. Nonuniqueness means that many different models (approximative solutions) could fit the observed data with the same accuracy (e.g., Tarantola 1987). The matter pertinent to the aforementioned parameter combination will be discussed further in the “Discussion” section.
Formulation of this 2.5D inverse problem and its solving
where G is the forward modeling operator acting on m to produce some predicted gravity data (g(x, A, z, Y, L, θ)) at a finite number (N) of observation points along a profile, m is a column vector of the model parameters (that is the vector of some approximative solution) we seek to retrieve from inversion (m=[A,z,Y,L,θ]^{ T }), g _{∘} is a column vector of a finite set of noisy gravity data measured along this profile, \(\phantom {\dot {i}\!}{g}_{\circ }=\left [g_{{\circ }_{1}}, g_{{\circ }_{2}}, g_{{\circ }_{3}}, \ldots g_{{\circ }_{N}}\right ]^{T}\), and T is the transposition operator.
Recent advances to solve the inverse problem of gravimetric data are based on the use of deterministic approaches that utilize the regularized least squares techniques and the differentiability of the objective function subject to minimization (e.g., Zhdanov 2002). As indicated earlier, the inversion scheme described in this paper recovers the characteristic parameters of the buried target in a minimal time (a few seconds). Therefore, automatic deterministic approaches are much more efficient (see, for example, Mehanee 2015) than the approaches that are based on the trialanderror modeling method.
where g is a column vector of the predicted data calculated along the profile, using formula (1), from some approximative solution m, g=[g _{1},g _{2},g _{3},…g _{ N }]^{ T }, and δ is the noise level embedded in the measured data (g _{ o }).
where \(C = \text {diag}\left [\sqrt {C_{z}}, \; \sqrt {C_{\theta }}, \; \sqrt {C_{L}}, \; \sqrt {C_{A}}, \; \sqrt {C_{Y}}\right ]\).
The first term of (6) is the data misfit functional, determined as the square norm of the difference between the observed and predicted data, and the second term is a stabilizing functional, the stabilizer.
where τ is a scaling parameter used in order to make Ψ dimensionless and is set to 1 \(\left (\tau = 1 \; \frac {1}{{\text {mGal}}^{\text {2}}}\right)\), and \(\widetilde {m} = [\log (A/A_{\circ }), \log (z/z_{\circ }), \log (Y/Y_{\circ }), \log (L/L_{\circ }), \log (\theta /\theta _{\circ }) ]^{T}\). Each of the parameters A _{∘}, z _{∘}, Y _{∘}, L _{∘}, and θ _{∘} (the socalled scaling parameters) is of a unit (positive) value and is introduced in order to make the quantities that are under the logs in \(\widetilde {m}\) dimensionless. The introduction of these parameters is physically necessary because the logs do not allow dimensions to be defined. As mentioned above, α is a dimensionless quantity. Note that hereinafter the scaling parameters of \(\widetilde {m}\) are dropped to simplify the notations.
All the numerical models and real data examples shown here are inverted and analyzed by the logarithmic minimization formulation presented in (7). We note that, in the restricted case of the noisefree and neighboring/interference effectfree numerical example (Model 1), the nonlinear minimization problem (7) is solved iteratively using a sequential hybrid technique that automatically combines the SD and GN methods (see Mehanee 2015, and the references therein) in order to verify and validate the developed inversion scheme prior to applying it to noisy and real data sets.
The use of the hybrid technique was essential. This is because the SD method converges very slowly or stagnates (as found from extensive noisefree numerical experiments) at the final stage of its iterative minimization process. It was also found from these experiments that the GN method essentially requires a very good initial guess in order to converge (Zhdanov 2002). That is why the SD method (which does not require a good initial guess to converge as will be seen in the “Numerical tests” subsection) is employed first in the developed scheme in order to generate and prepare a suitable initial guess for the GN method.
In the framework of this hybrid scheme, the SD method is used first until a normalized misfit (defined as \(\frac {\left \Vert \widetilde G(\widetilde {m}) \;  \; {g}_{\circ }\right \Vert }{\left \Vert {g}_{\circ }\right \Vert } \times 100~\% \)) of roughly around 5−10 % (depending on the model subject to inversion) is reached. After that, the GN method is applied to the observed data, where the inverse results produced from the SD method is used as the initial guess for the GN method. The GN method terminates when a normalized misfit below 10^{5} % is reached.
We note that when inverting the distorted (by noise and/or by interference due to nearby bodies) data sets (Model 2 and Model 3) shown in the “Numerical tests” subsection, and when inverting the two real data sets presented in the “Field data inversion” subsection, the minimization problem (7) was solely solved by the SD method. Unlike the noisefree data set (Model 1), we found in our experience for this particular inverse problem that the SD method can be sufficient for the inversion of the distorted and real field data examples. This is because in these examples (the distorted and real field data), we neither expect nor seek to exactly fit the observed data (see the figures pertinent to Model 2 and Model 3).
The entire computational steps of the GN and SD methods of the inverse scheme are presented and discussed in Appendices 1 and 2. The code of the scheme is implemented in MATLAB. We note that no builtin MATLAB optimization (minimization) functions were used in the 2.5D code of our paper.
Sensitivity (Fréchet) matrix calculation
The use of the logarithmed model parameters has the benefit of making the aforementioned derivatives dimensionally the same, which is a good strategy for balancing the Jacobian terms. The quantities \(\frac {\partial g}{\partial A}\), \(\frac {\partial g}{\partial L}\), \(\frac {\partial g}{\partial z}\), \(\frac {\partial g}{\partial \theta }\), and \(\frac {\partial g}{\partial Y}\) are evaluated analytically by differentiating the forward modeling formula (1).
Singular value decomposition
where U=[u _{1},u _{2},…,u _{ m }] ε R ^{ m×m } and V= [v _{1},v _{2},…,v _{ n }] ε R ^{ n×n } are both orthogonal matrices. S ε R ^{ m×n } is a diagonal matrix with nonnegative real values (the socalled singular values) arranged in decreasing order, s _{1}≥s _{2}……≥s _{ n }≥0. The sequence S=[s _{1},s _{2},…,s _{ n }] is referred to as the singular spectrum of \(\widetilde {{F}}\). The columns of U=[u _{1},u _{2},…,u _{ m }] and V=[v _{1},v _{2},…,v _{ n }] are the left and right singular vectors in the input and output spaces of the transformation represented by \(\widetilde {{F}}\), respectively. The magnitude of the singular values in S represents and determines the corresponding important singular vectors in the columns of U and V. The SVD is a useful tool for understanding the sensitivity analysis of the various model parameters of the 3D thinsheet model as will be seen.
Results and discussion
Numerical tests
Prior to applying the developed inversion algorithm to real data examples, its accuracy is assessed and analyzed first on three numerical models. All inverse solutions are rounded to the first integer.
Model 1: noisefree example
The presented algorithm supplemented with the SD method was applied to this noisefree dataset for 6000 iterations (yielding to a normalized misfit of about 4.7 %) using the initial guess shown at the top panel of Fig. 2. The corresponding inversion results (the socalled the preliminary inverse solution) are shown at the top panel of Fig. 2. To avoid possible stagnation with the SD method, the same algorithm supplemented with the GN method was applied to the same residual gravity data. The aforementioned preliminary inverse solution evolved from the SD method was employed automatically, as initial guess, in the GN method. The bottom panel of Fig. 2 shows the final inverse solution obtained from the scheme.
Model 2: noisy example
In order to assess and analyze the robustness and stability of the inverse algorithm, the developed scheme has been applied to a noisy profile. Subsequent noise levels of about 7, 11, and 20 % have corrupted the gravity data generated by a numerical model described by the following: z = 12 m, L = 35 m, Y = 100 m, θ=120°, and A = 12,000 kg/m ^{2}. Each noise level was separately generated and added to the noisefree data set \(\widetilde G(\widetilde {m})\) using the MATLAB function “awgn” in order to produce the corresponding corrupted data set g _{∘} subject to inversion. The aforementioned noise levels were separately calculated as \(\frac {\left \Vert {g}_{\circ } \;  \; \widetilde G(\widetilde {m}) \right \Vert }{\left \Vert {g}_{\circ }\right \Vert } \times 100\,\%\).
Model 2 (noisy data). Inversion results of various noise levels. The true model parameters are z = 12 m, L = 35 m, Y = 100 m, A = 12,000 kg/m ^{2}, and θ= 120 °
Noise level (%)  Inverse parameters  

z (m)  L (m)  Y (m)  A (kg/m ^{2})  θ (degrees)  
7  9.8  44.7  82  9945.5  115 
11  10  43.3  85  10,190  115.4 
20  10  48.6  80  9635  116.7 
Model 2 (noisy data). Error of the inversion results shown in Table 1
Noise level (%)  Error of inverse parameters (%)  

z  L  Y  A  θ  
7  18.3  27.7  18  17  4.2 
11  16.6  23.7  15  15  3.8 
20  16.6  38.9  20  19.7  2.8 
Model 3: interference effect example
Two interpretive scenarios are explored and investigated for analyzing and inverting the profile of the composite effect (Fig. 6, middle panel) by a single thin sheet (the subject of this paper). Scenario 1 (whole profile inversion) in which the whole profile is inverted and analyzed. Scenario 2 (extracted (truncated) profile inversion) in which only the distorted prominent anomaly (marked by arrows in Fig. 6, middle panel) out of the whole profile is inverted. Scenario 2, in this particular case, is probably more realistic (in terms of inverting its data by a single thinsheet model) as its gravity response can resemble a single thinsheet model. Scenario 1 is not so realistic (in terms of inverting its data by a single thinsheet model) as its gravity response subject to inversion has three anomalous signatures that cannot be attributed to or represented by a single thinsheet model. However, the inversion of scenario 1 has been carried out here solely for the sake of investigative purpose, better understanding and for the full illustration of the developed inversion scheme.
The top panel of Fig. 7 shows the true model parameters of the main body and the corresponding approximative inverse results of Scenario 1 for which a normalized misfit of about 10.85 % is reached using an α of 10^{−6}. The inversion results of Scenario 2 are shown in the bottom panel of Fig. 7. A normalized misfit of 6.71 % is obtained, and an α of 10^{−9} was used in the inverse computations. This analysis shows that the parameters A, L, and Y of the main body exhibited significant inaccuracy.
On the basis of this investigation, we can conclude that the developed scheme produces more accurate inversion results for an isolated target, and thus it is more suitable for isolated anomalies or for anomalies with insignificant neighboring/interference effect. Isolated targets (such as ore veins and dikes) are frequent in many geologic settings and can be of a paramount economic interest as will be seen in the next two real data examples in which the gravity data were acquired for chromite, copper, and gold prospecting.
Field data inversion
In order to examine and assess the applicability of the developed inversion technique, two published field examples, from mineral exploration in Canada and Cuba, with various depth of burial, geologic complexity, and interference effects are analyzed. These particular data sets were selected in this research paper for a number of reasons. First, these data sets were generated by buried causative ore bodies, which can resemble thinsheet models (Davis et al. 1957; Grant and West 1965; Siegel et al. 1957), the subject of this paper. Second, these data sets were measured from sites with known drilling information; hence, we can compare the numerical results yielded from the inversion against those confirmed from drilling. Third, these sites have core samples from which the density contrasts were accurately estimated in laboratory. As pointed out earlier, the knowledge of the density contrast (Δ ρ) is essential in order to compute the thickness (t) of the body from the amplitude coefficient (A) obtained from the inversion.
The Mobrun anomaly, Canada
Grant and West (1965) reported that the average density of core samples of the mineral taken from drilling was 4600 ± 500 kg/m ^{3}, and for the host rock, the density was about 2700 kg/m ^{3}. Hence, the ore body has a density contrast (Δ ρ) of 1400–2400 kg/m ^{3}. Thus, the aforementioned two density values are equal to the bulk densities of the ore and the host rock, respectively.
The inverse algorithm has been applied to the above mentioned residual gravity profile. Figure 8 shows the two approximative inverse solution sets obtained from two different regularization parameters (α) using an initial guess of (z = 2 m, L = 30 m, A = 2200 kg/m ^{2}, Y = 50 m, and θ=30°). The inversion results shown at the top panel were recovered using an α of 10^{−7} and correspond to a normalized misfit of 2.93 %. The bottom panel illustrates the results retrieved from the algorithm using an α of 10^{−4}, for which a normalized misfit of 2.48 % was reached. It is noted that the density contrast range mentioned above was used to estimate the corresponding thickness (t) range of the ore body from the amplitude coefficient (A=t Δ ρ) evolved from the inversion.
The Mobrun anomaly, Canada. Tabulated inversion results. α is the regularization parameter
Misfit %  α  Initial guess  Inverse solution  

z (m)  L (m)  Y (m)  A (kg/m ^{2})  θ (degrees)  z (m)  L (m)  Y (m)  A (kg/m ^{2})  θ (degrees)  
2.93  10^{7}  2  30  50  2200  30  11  203  125  52,604  86 
2.48  10^{4}  2  30  50  2200  30  14  172  113  58,801  86 
2.7  10^{2}  10  500  500  2000  120  12  178  118  55,968  86 
2.47  10^{4}  10  500  500  2000  120  14  119  262  59,631  86 
2.7  10^{4}  10  1000  80  20,000  10  17  299  55  75,420  86 
2.67  10^{3}  10  1000  80  20,000  10  16  312  59  71,188  86 
One can see that the two approximative solution sets (the socalled here “Equivalent Model 1” and “Equivalent Model 2”) shown at rows 3 and 5 of Table 3 are different, though each of them has the same normalized misfit value (2.7 %). This is attributed to the nonuniqueness nature of this particular inverse problem; that is, different inverse models can equally fit the observed data.
The parameter A retrieved from inversion (A = 55,791.2 kg/m ^{2}) suggests a thickness of 23–40 m (t = 23–40 m) which is greater than the depth recovered from inversion (z=12.4 m). Probably this is because the actual ore body is not a thin sheet. Supporting evidence is that the actual ore body has a thickness of 31 m as revealed from a slice (constructed from drilling) taken at 50 m depth (Fig. 9). And that the depth to the top of the target is less than 30.5 m as inferred from directional drilling that intersected the body at about 30.5 m. As pointed out in the “Forward modeling solution” subsection, the 3D thin sheet is a good approximation to a prismatic structure, unless the thickness (t) is somewhat greater than the depth to the top (z) (Telford et al. 1976). However, we sought in this paper to fit and interpret this residual gravity profile by a thinsheetlike model for investigative purposes and the full illustration of the developed inverse scheme, and in order to gain some insights.
Prior to any drilling in this area, Siegel et al. (1957) analyzed this gravity profile based on a 2D trial and error modeling method and reported a corresponding solution of z=6 m, L=186 m, and t=31 m. It is relevant to note that the strike length of the deposit in this case is assumed to be infinite as the interpretation was essentially based on 2D assumption. Grant and West (1965) too interpreted this profile but as a 3D thin sheet, based on a graphical method, and the retrieved inverse solution is as follows: z=17 m, L=170 m, 2 Y (strike length) = 205 m, θ=83°, and t=36 m. These results, too, show that the approximative model is quasivertical. Using an Euler deconvolution technique, Roy et al. (2000) interpreted this profile by a 2D vertical ribbon model and obtained an inverse model of z = 22.7 m and L = 52 m. While their results are in some variations from the true results confirmed from drilling, these results are still useful as they could be used as a reasonable initial guess in deterministic (gradienttype) inversions. As noted above, other gravity inversion methods (e.g., Li and Oldenburg 1998; Zhdanov et al. 2004) could produce a more accurate inverse solution and fitting.
Sensitivity analysis
In order to get some insights on a possible mutual interrelation between the nonuniqueness of this particular inverse problem and the model parameter sensitivities (that is, the Fréchet matrix; \(\widetilde {{F}} = \frac {\partial g}{\partial \widetilde {m}}\)), the sensitivities of the aforementioned two equivalent solutions (“Equivalent Model 1” and “Equivalent Model 2”) and of the two initial guesses used in the inversion of these two equivalent solutions have been calculated.
Note that the sensitivities of the inverse parameters log(L) and log(Y) (the two main parameters that appear to be most affected by the nonuniqueness of this particular inverse problem as will be strengthened further and seen in the sensitivity subsection of the field example of the Camaguey area, Cuba) of “Equivalent Model 1” and “Equivalent Model 2” are found to be positive real numbers. The bottom panel (corresponding to these two equivalent models) of Figs. 10 and 11 qualitatively reveals that the sensitivity curves of these two parameters (log(L) and log(Y)) have a similar behavior and form. The degree of conformity of these two sensitivity curves depends upon the values of the model parameters used in the sensitivity calculation. The top and bottom panels of Figs. 10 and 11 show that the parameter log(θ) has the dominant sensitivity. Pertinent quantitative analysis based on the singular value decomposition (e.g., Press et al. 1988) is provided in the next paragraph.
In order to understand better the mutual interrelation between the nonuniqueness of this particular inverse problem and the model parameters sensitivity, the SVD (e.g., Jupp and Vozoff 1975, Press et al. 1988) was carried out separately on the Fréchet matrix of “Equivalent Model 1” and of the initial guess of this equivalent model.
The Mobrun anomaly, Canada. Singular values and their variabilities in percentage of the sensitivity matrix, which was calculated from the initial guess shown at the top panel of Fig. 10
Singular values  Variability in percentage 

19.7  97.73 
0.394  1.96 
0.044  0.22 
0.0171  0.085 
0.0019  0.0094 
The Mobrun anomaly, Canada. Singular values and their variabilities in percentage of the sensitivity matrix, which was obtained from “Equivalent Model 1” shown at the bottom panel of Fig. 10
Singular values  Variability in percentage 

244.78  96.76 
6.78  2.68 
1.06  0.42 
0.33  0.13 
0.03  0.013 
The Camaguey Province anomaly, Cuba
Inverse results
Davis et al. (1957) carried out detailed gravity surveys in the Camaguey Province, Cuba, for chromite exploration. Residual gravity maps were collected over various spatially distributed ore bodies for determining approximately the quantity of chromite in any deposits found. In this paper, a profile from this area is inverted and interpreted. This profile is taken normal to the strike of the residual gravity anomaly shown in the middle part of Fig. 6 of Davis et al. (1957).
This gravity profile is associated with the largest chromite deposit found in the province (Davis et al. 1957). This prominent profile has a trend of S60 ^{ o }W to N60 ^{ o }E and overlies a chromite ore body which contains about 115,000 tons, dips steeply to the southwest, and comes within 3 m of the ground surface (Davis et al. 1957).
The Camaguey Province anomaly, Cuba. Tabulated inversion results
Misfit %  α  Initial guess  Inverse solution  

z (m)  L (m)  Y (m)  A (kg/m ^{2})  θ (degrees)  z (m)  L (m)  Y (m)  A (kg/m ^{2})  θ (degrees)  
7.56  10^{4}  100  200  300  20,000  30  6  40  30  6990  94 
7.56  10^{6}  50  350  300  10,000  120  6  30  42  7647  94 
7.56  10^{6}  75  20  20  15,000  10  7  30  36  7980  94 
The sensitivity of the inverse parameters log(L), log(Y), and log(A) of “Equivalent Model 1” and “Equivalent Model 2” are found positive real number for this profile. The bottom panels of Figs. 15 and 16 show that these three parameters have quasisimilar behavior and form—this is consistent with the pertinent findings reported for the Mobrun anomaly, Canada (see the bottom panels of Figs. 10 and 11).
Sensitivity analysis
The Camaguey Province anomaly, Cuba. Singular values and their variabilities in percentage of the sensitivity matrix, which was calculated from the initial guess shown at the top panel of Fig. 15
Singular values  Variability in percentage 

8.22  79.1 
2.12  20.4 
0.049  0.47 
0.0006  0.0055 
0.0002  0.0020 
The Camaguey Province anomaly, Cuba. Singular values and their variabilities in percentage of the sensitivity matrix, which was calculated from “Equivalent Model 1” shown at the bottom panel of Fig. 15
Singular values  Variability in percentage 

33.66  96.88 
0.861  2.48 
0.18  0.52 
0.038  0.111 
0.0033  0.009 
The SVD analysis presented herein revealed that the parameter log(θ) has the highest importance in this inverse scheme. All field examples, carefully analyzed here, have shown that the parameter θ is determined very accurately (based on drilling information), and that this parameter is the least affected by the initial guess selection and the misfit stopping criteria. It is worthy to note that accurate determination of the amount and direction of dip (θ) of the buried target can be of a paramount importance for effective decision making on directional drilling.
On the basis of the inversion results of the two investigated field examples, it can be concluded that the model parameter which can be estimated from the developed method with the greatest accuracy is θ. The parameters z, A, L, and Y can be determined with an acceptable accuracy, especially when the gravity response is due to an isolated body.
Discussion
Gravity inversions based on regular models remain of interest in exploration geophysics (see for example, Biswas 2015, and the references therein). It is worthy to note that it is very rare to find a geologic target that is truly regular body. Nevertheless, these regular models often yield a first approximation solution, which is sometimes quite adequate (e.g., Abdelrahman et al. 1989; Mehanee 2014a).

For reconnaissance analysis and pilot studies (for first approximative interpretation) in geophysical prospecting prior to conducting largescale gravity data surveys,

For comparative and ambiguity studies and joint (e.g., gravity and magnetic data) inversion in order to minimize the nonuniqueness that is imperative here, and

For integrated and unequivocal interpretation. For example, this scheme can determine the amount and direction of dip of the buried target with greatest accuracy (as seen in the investigated inversions of the noisy numerical data and the field examples), which was confirmed by the sensitivity analysis carried out in our paper.
This scheme is complimentary not contradictory to the existing interpretation methods. For example, the approximative inverse solution evolved from the thinsheet inverse scheme can be used to build a reasonable initial guess for the rigorous 3D gravity inverse schemes. Note that a reasonable initial guess is a prerequisite for the rigorous 3D gravity inverse schemes. It is emphasized that we are not claiming that inversion schemes based on geometrically simple bodies replace the rigorous 3D gravity inverse schemes.
Rigorous 3D gravity inversion schemes (e.g., Li and Oldenburg 1998; Zhdanov et al. 2004) can simultaneously invert for several anomalous irregular bodies and normally require areal data coverage. However, these schemes take much longer computation time (than the thinsheet schemes) in order to yield the approximative inverse solution (that is also nonunique), which comprises the 3D anomalous density distribution (Δ ρ) in the subsurface. As indicated above, a reasonable initial guess is a prerequisite in the rigorous 3D gravity inverse schemes. These rigorous 3D schemes can generate smooth or focused (compact) inverse images (depending primarily upon the particular objective of the interpretation) by incorporating the appropriate type of the stabilizer in the objective functional subject to minimization. It is relevant to note that the inversion of focused (compact) images requires accurate information about the lower and upper bounds of the anomalous density distribution (Δ ρ) of the buried targets.
Upon the completion of the gravity data acquisition and processing for an area, the following interpretation steps (e.g., Grant and West 1965; Hinze et al. 2013) are applied in order to estimate the characteristic parameters of a buried target:
First, contour the measured residual gravity data (regardless whether these data were acquired in a disordered or an ordered pattern). Second, examine the obtained contour map, and identify the strike of the buried target (Hinze et al. 2013). Third, take a profile traversing the center of the target and normal to the target’s strike direction (e.g., Hinze et al. 2013). Fourth, under the thinsheet assumption, invert this profile by the 2.5D scheme developed here using several initial guesses. This will give first approximate interpretation about the underlying body. The value of the parameter θ (which determines the amount and direction of dip of the buried body) will be the most accurate (see column 12 of Tables 3 and 6) as confirmed from the numerical models, filed examples, and the sensitivity analysis. The obtained inverse solutions should be interpreted in an integrated manner with all available geological and geophysical information.
Grouping the density contrast (Δ ρ) and thickness (t) of a 3D dipping thinsheet body into a single inverse parameter A (A=Δ ρ t) was essentially carried out because both of them appear only as a multiplicative combination. This grouping was done in order to help minimize the nonuniqueness nature of this particular inverse problem. Unfortunately, the rest of the inverse parameters (z,L,Y, and θ) we seek to recover from inversion are present frequently at various terms within formula (1) and are multiplicative (that is, they are coupled). Therefore, grouping any of these parameters (z,L,Y, and θ) is not really helpful in eliminating or mitigating the nonuniqueness of this inverse problem.
The particular 2.5D inverse problem developed here is nonunique as has been seen, thus it is illposed. A way of solving an illposed problem is via the use of regularization (Gribok et al. 2002; Tikhonov and Arsenin 1977). Regularization is required here for a number of reasons: First, in order to help the minimizer combat the entrapment in a local minimum. This can be accomplished by attempting various values for the regularization parameter in the scheme when inverting a gravity data set using an initial guess (see, for example, Fig. 8 and Table 3). Second, in order to find possible equivalent approximative solutions to understand better the nonuniqueness nature of this particular inverse problem, and thus to interpret these solutions in an integrated manner with the available geological, geophysical, and drilling data. Third, in order to make it possible to incorporate some a priori information, if available, in the stabilizer of the objective functional subject to minimization, which can help minimize the nonuniqueness nature of this inverse problem. The regularization parameter α is chosen such that it makes some balance between the misfit functional term and stabilizer (see, e.g., Li and Oldenburg 2003; Mehanee 2014b, and the references therein).
The inherent nonuniqueness of this inverse problem and the errors in the residual gravity data lead to the existence of equivalent solutions. If approximate prior knowledge on one of the target’s parameters (e.g., extension in depth L) is known, then we can incorporate this a priori information in the stabilizer of the objective functional. This could reduce the equivalence problem. It is not mere coincidence that the results obtained here for the two investigated field data examples have been found in some conformity with those confirmed by drilling. Therefore, the inverse scheme developed here can have great potential in exploration and mining geophysics for an isolated target described by a 3D thinsheet type model.
Conclusions
We have developed an efficient regularized iterative 2.5D inversion scheme for the interpretation of a residual gravity profile measured over a dipping thinsheet like target. The scheme determines the characteristic parameters (depth to top z, amount and direction of dip θ, extension in depth L, finite strike extension 2 Y, and amplitude coefficient A from which the thickness t is obtained) of a buried target. The algorithm of the scheme solves for the inverse parameters of a model in the space of their logarithms. This is advantageous for a number of reasons: First, in order to maintain the convergence of the objective functional subject to minimization. Second, in order to impose the positivity of the model parameters we seek, and hence, to produce realistic and meaningful inversion results. Third, in order to balance the Jacobian terms and to make the sensitivity derivatives dimensionally the same. It has been shown that this new scheme is advantageous in terms of computational efficiency, stability, and convergence than the existing schemes that solve for the characteristic inverse parameters of a sheet/dike model from gravity data inversions.
Before applying the method to real data examples, it has been successfully verified on noisefree numerical examples; it has recovered the actual model parameters. After that, the approach was assessed on noisy numerical data, and it is found stable and can estimate the parameters of the buried deposit with acceptable accuracy. However, some of the inverse parameters encountered some inaccuracy when the method was applied to residual data distorted, in terms of both magnitude and shape, by some significant neighboring gravity effects generated by nearby anomalous bodies.
The validity of the technique for practical applications has been successfully illustrated on two real field examples with various geologic settings and complexities from mineral exploration. The estimated inverse parameters of the investigated real data are found to generally conform well with those yielded from drilling.
The sensitivity analysis carried out on the Jacobian matrices of the field examples investigated here has shown that the parameter that can be determined with the greatest accuracy is θ (as confirmed from drilling information). Accurate determination of the amount and direction of dip (θ) of the buried target can be of a paramount importance for effective decisionmaking on directional drilling. The parameters z, L, Y, and A can be estimated with acceptable accuracy, especially the parameters z and A. Real data inversions have shown that the parameter θ is the least affected by the choice of the initial guess and the misfit stopping criteria.
The developed inversion scheme is useful in exploration programs and reconnaissance studies intended to delineate and map, for example, ore veins from residual gravity data. However, it can produce a nonunique inverse solution; a fact which should be kept in mind when interpreting the obtained approximative inverse solutions. Therefore, these equivalent solutions, as always, should be guided by geological information and other available geophysical results to help resolve any encountered nonuniqueness, which is not unusual in exploration geophysics. The nonuniqueness analysis and the tabulated inverse results presented here have shown that the parameters that are most affected by the nonuniqueness, the choice of the initial guess, and the misfit value of the stopping criteria are L and Y.
Simultaneous inversion of 3D dipping thinsheetlike multibodies will be the subject of future research.
The gravity signature due to a buried target depends upon the density contrast among the other characteristic parameters (e.g., extension in depth and extension in the strike direction) of the buried target. The density contrast between the ore body and the country rock can be in some cases much smaller than the magnetic susceptibility contrast. Therefore, the buried target in this case may generate a more prominent magnetic signature than the gravimetric one. Thus, the magnetic method, in this case, could be of a more value than the gravity method. Therefore, the 2.5D inversion scheme developed here could also be extended and useful in the interpretation of magnetic data.
Appendix
Appendix 1 GaussNewton (GN) method
where \(\widetilde {{ m}}_{n}\) is the column vector of the model parameters at iteration n; \(\widetilde {{ m}}_{n} = [\log (A_{n})\), log(z _{ n }), log(Y _{ n }), log(L _{ n }), log(θ _{ n })]^{ T }; T is the transpose operator; R _{ n } is the column vector of the difference between the predicted (G(m _{ n })) and observed (g _{∘}) gravity data sets at iteration n; \(\widetilde {{F}}_{n}\) is the Fréchet (Jacobian) matrix (Menke 2012; Tarantola 1987, 2005) computed at iteration n with respect to the log of the model parameters; α is the regularization parameter; \(\widetilde {\delta {m}_{n}}\) is the model parameters update at iteration n; \(\widetilde {{{l}}}_{n}\) is the direction of the steepest ascent computed at iteration n; and I and H are the identity and Hessian matrices.
Appendix 2: Steepest descent (SD) method
Declarations
Acknowledgments
We wish to thank Professor Yasuo Ogawa, editorinchief, and assistant Professor Yosuke Aoki, associate editor, for their time and efforts that led to the conciseness of the paper. We also wish to thank four anonymous reviewers for their helpful and constructive comments and inputs. One of the authors (Mehanee) wishes to thank Dr. Birendra Pandey for making some papers available.
Authors’ Affiliations
References
 Abdelrahman, EM, Bayoumi AI, Abdelhady YE, Gobashy MM, Elaraby HM (1989) Gravity interpretation using correlation factors between successive leastsquares residual anomalies. Geophysics 54: 1614–1621.View ArticleGoogle Scholar
 Ateya, IL, Takemoto S (2002) Gravity inversion modeling across a 2D dikelike structure—a case study. Earth Planets Space 54: 791–796.View ArticleGoogle Scholar
 BatistaRodríguez, JA, PérezFlores MA, UrrutiaFucugauchi J (2013) Threedimensional gravity modeling of Chicxulub Crater structure, constrained with marine seismic data and land boreholes. Earth, Planets and Space 65: 973–983.View ArticleGoogle Scholar
 Beiki, M, Pedersen LB (2011) Window constrained inversion of gravity gradient tensor data using dike and contact models. Geophysics 76: 159–172.Google Scholar
 Biswas, A (2015) Interpretation of residual gravity anomaly caused by simple shaped bodies using very fast simulated annealing global optimization. Geosci Frontiers. doi:http://dx.doi.org/10.1016/j.gsf.2015.03.00110.1016/j.gsf.2015.03.001.
 Davis, WE, Jackson WH, Richter DH (1957) Gravity prospecting for chromite deposits in Camaguey Province, Cuba. GeophysicsXXII(4): 848–869.View ArticleGoogle Scholar
 Fedi, M (2007) DEXP: a fast method to determine the depth and the structural index of potential field source. Geophysics 72: 1–11.View ArticleGoogle Scholar
 Grant, FS, West GF (1965) Interpretation theory in applied geophysics. McGrawHill Book Co, New York.Google Scholar
 Gribok, A, Hines JW, Urmanov A, Uhrig RE (2002) Heuristic, systematic, and informational regularization for process monitoring. Int J Intell Syst 17: 723–749.View ArticleGoogle Scholar
 Hinze, WJ (1990) The role of gravity and magnetic methods in engineering and environmental studies. In: Ward S. H. (ed)Geotechnical and environmental geophysics, Vol I: review and tutorial. Society of Exploration Geophysicists.. OK, Tulsa.Google Scholar
 Hinze, WJ, von Frese RRB, Saad AH (2013) Gravity and magnetic exploration: principles, practices and applications In: Cambridge University Press.. USA, New York.Google Scholar
 Holstein, H, Ketteridge B (1996) Gravimetric analysis of uniform polyhedra. Geophysics 61: 357–364.View ArticleGoogle Scholar
 Holstein, H (2003) Gravimagnetic anomaly formulas for polyhedra of spatially linear media. Geophysics 68: 157–167.View ArticleGoogle Scholar
 Holstein, H, Fitzgerald D, Anastasiades C (2010) Gravimagnetic anomalies of uniform thin polygonal sheets In: 72nd European Association of Geoscientists and Engineers conference &; exhibition incorporating SPE EUROPEC, Barcelona, Spain, 14–17 June.Google Scholar
 Holstein, H, Anastasiades C (2010) Finitely expanded gravity anomaly of uniform thin polygonal sheets In: 72nd European Association of Geoscientists and Engineers conference & exhibition incorporating SPE EUROPEC, Barcelona, Spain, 14–17 June.Google Scholar
 Holstein, H, Hillan D, Fitzgerald D (2014) Gravity anomalies of polygonal sheets with linear density variation In: 76th European Association of Geoscientists and Engineers conference & exhibition, Amsterdam RAI, The Netherlands, 16–19 June.Google Scholar
 Jupp, DLB, Vozoff K (1975) Stable iterative methods for the inversion of geophysical data. Ceophys J R Astr Soc 42: 957–976.View ArticleGoogle Scholar
 LaFehr, TR (2012) Fundamentals of gravity exploration. Society of Exploration Geophysicists, Tulsa, Ok, USA.Google Scholar
 Levenberg, K (1944) A method for the solution of certain problems in least squares. Quart Appl Math 2: 164–168.Google Scholar
 Li, Y, Oldenburg DW (1998) 3D inversion of gravity data. Geophysics 63: 109–119.View ArticleGoogle Scholar
 Li, Y, Oldenburg DW (2003) Fast inversion of largescale magnetic data using wavelet transforms and a logarithmic barrier method. Geophys J Int 152: 251–265.View ArticleGoogle Scholar
 Long, LT, Kaufmann RD (2013) Acquisition and analysis of terrestrial gravity data. Cambridge University Press, New York, USA.View ArticleGoogle Scholar
 Madsen, K (2004) Methods for nonlinear least squares problems. Informatics and Mathematical Modelling, Technical University of Denmark Press, Denmark.Google Scholar
 Marquardt, D (1963) An algorithm for least squares estimation on nonlinear parameters. SIAM J Appl Math 11: 431–441.View ArticleGoogle Scholar
 Mehanee, S, Zhdanov M (2002) Twodimensional magnetotelluric inversion of blocky geoelectrical structures. J Geophys ResSolid Earth 107: 191–202.View ArticleGoogle Scholar
 Mehanee, S (2014a) Accurate and efficient regularized inversion approach for the interpretation of isolated gravity anomalies. Pure Appl Geophys 171: 1897–1937.View ArticleGoogle Scholar
 Mehanee, S (2014b) An efficient regularized inversion approach for selfpotential data interpretation of ore exploration using a mix of logarithmic and nonlogarithmic model parameters. Ore Geology Rev 57: 87–115.View ArticleGoogle Scholar
 Mehanee, S (2015) Tracing of paleoshear zones using selfpotential data inversion: case studies from the KTB, Rittsteig, and Grossensees graphitebearing fault planes. Earth Planets Space 67: 14–47.View ArticleGoogle Scholar
 Menke, W (2012) Geophysical data analysis: discrete inverse theory, Matlab Edition. 3rd Edition. Academic Press, New York.Google Scholar
 Mushayandebvu, MF, van Drielz P, Reid AB, Fairhead JD (2001) Magnetic source parameters of twodimensional structures using extended Euler deconvolution. Geophysics 66: 814–823.View ArticleGoogle Scholar
 Nettleton, LL (1976) Gravity and magnetics in oil prospecting. McGrawHill Book Co, New York.Google Scholar
 Okubo, S, Tanaka Y, Ueki S, Oshima H, Maekawa T, Imanishi Y (2013) Gravity variation around Shinmoedake volcano from February 2011 through March 2012—results of continuous absolute gravity observation and repeated hybrid gravity measurements. Earth, Planets Space 65: 563–571.View ArticleGoogle Scholar
 Paoletti, V, Ialongo S, Florio G, Fedi M, Cella F (2013) Selfconstrained inversion of potential fields. Geophys J Int 195: 854–869.View ArticleGoogle Scholar
 Pei, J, Li H, Wang H, Si J, Sun Z, Zhou Z (2014) Magnetic properties of the Wenchuan Earthquake Fault Scientific Drilling Project Hole1 (WFSD1), Sichuan Province, China. Earth Planets Space 66: 23.View ArticleGoogle Scholar
 Pirajno, F (2013) GonzálezÁlvarez I. A reappraisal of the epoch nickel sulphide deposit, Filabusi Greenstone Belt, Zimbabwe: a hydrothermal nickel mineral system?. Ore Geology Rev 52: 58–65.Google Scholar
 Press, WH, Flannery BP, Teukolsky SA, Vetterling WT (1988) Numerical recipes in FORTRAN. Cambridge University Press, Cambridge.Google Scholar
 Ramlau, R (2005) On the use of fixed point iterations for the regularization of nonlinear illposed problems. J Inv IllPosed Problems 13: 175–200.Google Scholar
 Roy, L, Agarwal BNP, Shaw RK (2000) A new concept in Euler deconvolution of isolated gravity anomalies. Geophys Prospecting 48: 559–575.View ArticleGoogle Scholar
 Santosh, M, Pirajno F (2014) Ore deposits in relation to solid earth dynamics and surface environment. preface. Ore Geology Rev 56: 373–375.View ArticleGoogle Scholar
 Sazhina, N, Grushinsky N (1971) Geophysical prospecting. MIR Publishers, Moscow.Google Scholar
 Siegel, HO, Winkler HA, Boniwell JB (1957) Discovery of the Mobrun Copper Ltd. sulphide deposit, Noranda Mining District, Quebec In: Methods and case histories in mining geophysics. Commonwealth Mining Met. Congr., 6th, 237–245, Vancouver.Google Scholar
 Tarantola, A (1987) Inverse problem theory. Elsevier, New York: 613.Google Scholar
 Tarantola, A (2005) Inverse problem theory and methods for model parameters estimation. Society of Industrial and Applied Mathematics (SIAM), Pennsylvania.View ArticleGoogle Scholar
 Telford, WM, Geldart LP (1976) Applied geophysics. Cambridge University Press, Cambridge.Google Scholar
 Tikhonov, AN, Arsenin VY (1977) Solutions of illposed problems. Wiley and Sons, New York.Google Scholar
 Tikhonov, AN, Leonov AS, Yagola AG (1998) Nonlinear illposed problems, Vols 1 and 2. Chapman and Hall, London.Google Scholar
 Zhdanov, MS (2002) Geophysical inversion theory and regularization problems. Elsevier, Amsterdam.Google Scholar
 Zhdanov, MS, Robert E, Mukherjee S (2004) Threedimensional regularized focusing inversion of gravity gradient tensor component data. Geophysics 69: 925–937.View ArticleGoogle Scholar
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