Fast 3D inversion of gravity data using Lanczos ...

Fast 3D inversion of gravity data using Lanczos bidiagonalization method

Reza Toushmalani & Hakim Saibi

Arabian Journal of Geosciences ISSN 1866-7511 Volume 8 Number 7 Arab J Geosci (2015) 8:4969-4981 DOI 10.1007/s12517-014-1534-4

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Author's personal copy Arab J Geosci (2015) 8:4969–4981 DOI 10.1007/s12517-014-1534-4

ORIGINAL PAPER

Fast 3D inversion of gravity data using Lanczos bidiagonalization method Reza Toushmalani & Hakim Saibi

Received: 29 January 2014 / Accepted: 30 June 2014 / Published online: 17 July 2014 # Saudi Society for Geosciences 2014

Abstract This paper describes the application of a new inversion method to recover a three-dimensional density model from measured gravity anomalies. To attain this, the survey area is divided into a large number of rectangular prisms in a mesh with unknown densities. The results show that the application of the Lanczos bidiagonalization algorithm in the inversion helps to solve a Tikhonov cost function in a short time. The performance time of the inverse modeling greatly decreases by substituting the forward operator matrix with a matrix of lower dimension. A least-squares QR (LSQR) method is applied to select the best value of a regularization parameter. A Euler deconvolution method was used to avoid the natural trend of gravity structures to concentrate at shallow depth. Finally, the newly developed method was applied to synthetic data to demonstrate its suitability and then to real data from the Bandar Charak region (Hormozgan, south Iran). The 3D gravity inversion results were able to detect the location of the known salt dome (density contrast of −0.2 g/cm3) intrusion in the study area.

Keywords Inverse problem . 3D modeling . Lanczos bidiagonalization . Gravity anomaly . Bandar Charak

R. Toushmalani Department of Computer, Faculty of Engineering, Kangavar Branch, Islamic Azad University, Kangavar, Iran H. Saibi (*) Laboratory of Exploration Geophysics, Department of Earth Resources Engineering, Faculty of Engineering, Kyushu University, 744 Motooka, Nishi-ku, Fukuoka 819-0395, Japan e-mail: [email protected]

Introduction Inversion is defined here as an automated numerical procedure used to construct a model of subsurface physical property (density) variations from measured data and any prior information independent of the measured data. Quantitative interpretation is then carried out by drawing geologic conclusions from the inverted models. A model is either parameterized to describe the source geometry or is described by a distribution of a physical property, such as density or magnetic susceptibility contrast. The development of inversion algorithms naturally follows these two directions. Bott (1959) first attempted to invert for basin depth from gravity data by adjusting the depth of vertical prisms through trial and error. Danes (1960) used a similar approach to determine the location of the top of a salt dome. Oldenburg (1974) adopted Parker’s (1973) forward procedure in the Fourier domain to formulate an inversion algorithm for basin depth by applying formal inverse theory. Green (1975) used an appropriate weighting matrix to fix some of the parameters when geologic or density information was available. Green (1975) applied the Backus-Gilbert approach to invert 2D gravity data and guided the inversion by using reference models and associated weights constructed from prior information. In a similar direction, Last and Kubik (1983) guided the inversion by minimizing the total volume of the causative body, and Guillen and Menichetti (1984) chose to minimize the inertia of the body with respect to the center of the body or an axis passing through it. In inverse problems, the measured data are used, together with a forward linear or nonlinear mapping operator, to find the model parameters that characterize some physical process (Menke 1989). A number of authors extended the approach to different density-depth functions or imposed various constraints on the basement relief (e.g., Barbosa et al. 1997). Although a solution that satisfies the observed data can easily be found, the solution

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is nonunique. The under-determination of the problem is further increased when considering data uncertainty. Many strategies can be used to address the nonuniqueness problem in gravity inversion. They all involve applying some type of constraints or regularization (Gallardo-Delgado et al. 2003) to limit the resulting solution space. Last and Kubik (1983) sought a compact solution with a minimum volume constraint. The smoothness or roughness of the density distribution controlling the gradients of the parameters in the spatial directions was used in a magnetic inversion by Pilkington (1997). Recently, this general methodology has been used extensively in inversions for the base of salt in oil and gas exploration (e.g., Cheng et al. 2003). A similar approach has been used to invert for the geometry of isolated causative bodies by representing them as polygonal bodies in two dimensions or as polyhedral bodies in three dimensions (Pedersen 1977; Moraes and Hansen 2001), in which the vertices of the objects are recovered as the unknowns. Alternatively, one may invert for density contrast as a function of the position in the subsurface. While these approaches are effective, they are limited in that they are only able to recover single bodies. Li and Oldenburg (1998) formulated a generalized 3D inversion of gravity data using the Tikhonov regularization and a model objective function that measures the structural complexity of the model. A lower and upper bound are also imposed on the recovered density contrast to further stabilize the solution. A similar approach has been extended to the inversion of gravity gradient data (Li 2001; Zhdanov et al. 2004). More recently, there have been efforts to combine the strengths of these two approaches. Krahenbuhl and Li (2002, 2004) formulated the base-of-salt inversion as a binary problem, and Zhang et al. (2004) took a similar approach for crustal studies. Interestingly, the genetic algorithm has been used as the basic solver in the latter approaches. This is an area of growing interest, especially when a refinement of the inversion is desired using constraints based on prior information. Full 3D density inversions have been devised and successfully applied to a variety of geologic settings; see, for example, Zhdanov et al. (2004) and Silva Dias et al. (2011). All these methods involve a discretization of the model space into 3D voxels, resulting in very large, but linear, optimization problems. By utilizing compression techniques, Portniaguine and Zhdanov (2002) showed how the computational burden can be reduced to a manageable size. Li and Oldenburg (2003) used wavelet transforms and compression to yield efficient matrix vector multiplications. Other methods to increase the efficiency include making the discretization adaptive so that the vertical boundaries of the voxels are moved to the depths where the property is changing. Spatial inversions for the surface defining the interface between geologic bodies have also been implemented and used on field data; see, for example, Barbosa et al. (1997).

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These methods minimize a non-linear function containing terms relating to the data misfit function and a regularizing function that imposes certain behavior on the surface. Li and Oldenburg (1998) counteracted the decreasing sensitivities of cells with increasing depth by weighting them with an inverse function of depth. Li and Oldenburg (2000) and Lelièvre and Oldenburg (2009) also investigated options for incorporating structural orientation information into underdetermined inversions in a deterministic framework (i.e., minimization of an objective function). Barbosa and Silva (1994) and Dias et al. (2009) also concentrated the solution along inertial axes. Stochastic inversion methods (e.g., Franklin 1970; Mosegaard and Tarantola 1995; Eidsvik et al. 2004; Tarantola 2005) have also proven useful for many inversion problems. Among the stochastic methods, geostatistical inversion algorithms are gaining popularity. They have been applied to geophysical data by Haas and Dubrule (1994); Torres-Verdin et al. (1999); Asli et al. (2000); Chasseriau and Chouteau (2003); Gloaguen et al. (2005); Bosch et al. (2006); Hansen et al. (2006); and Giroux et al. (2007). Additionally, Shamsipour et al. (2010, 2011, 2012) presented the geostatistical techniques of cokriging and conditional simulation for the separate and joint 3D inversion of gravity and magnetic data. The recent work of Higdon et al. (1999) popularized nonstationary covariance models based on convolutions. Paciorek and Schervish (2006) extended the work of Higdon et al. (1999) to create a class of closed-form nonstationary covariance functions, including the nonstationary Matern covariance model, parameterized by spatially varying parameters. Fuentes (2001) also proposed a kernelbased model in which the spatial field is obtained as a convolution of a stationary field. Shamsipour et al. (2013) adopted the idea of kernel convolution of Paciorek and Schervish (2006) to account for the spatial variation of the covariance function in the stochastic inversion of gravity data methods proposed by Shamsipour et al. (2010). Camacho et al. (2000, 2002) proposed a method of 3D gravity inversion that is inspired by the method of René (1986), which is based on a “growth” process and the mathematical exploration of the model space (in the sense of Tarantola 1988) for defined density contrasts. Camacho et al. (2011) presented the gravity inversion software GROWTH2.0 and its application to recently obtained gravity data from the volcanic island of Tenerife (Canary Islands, Spain), which provides information about the subsurface density structure. GROWTH2.0 is an inversion tool that enables the user to obtain, in a nearly automatic and nonsubjective mode, a 3D model of the subsurface density anomalies based on observed gravity anomaly data. Quantitative interpretation is then performed to illustrate geologic conclusions from the inverted models. A model is parameterized to explain the source geometry or is described

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by a distribution of a physical property, such as the density contrast (Nabighian et al. 2005). The main objective of this paper is to develop a new and fast 3D gravity inversion technique that is capable of quickly recovering 3D density distributions. We recommend the use of a new technique, which actually avoids large matrix multiplication. To achieve this, an iterative technique called Lanczos bidiagonalization is used. The original system of equations is substituted with a system of lesser dimension to increase the speed of the solution procedure significantly while being able to solve the original problem with a high degree of accuracy (Abedi et al. 2013). A least-squares QR (LSQR) method is applied to select the best value of a regularization parameter. In what follows, after a short introduction of the inverse procedure, the method is applied to synthetic data to show its suitability for 3D inversion. To this end, the new method was applied to real gravity data obtained from the Bandar Charak region located in Hormozgan province in south Iran. Mapped 3D density distributions in the study area show the known salt dome intrusion.

Regularized method Description of forward and inverse gravity problem Forward modeling Consider a data set consisting of N observations of anomalous gravity data. We suppose these observations are a result of density anomalies confined in a threedimensional region R of a given subsurface. To show these density anomalies as discrete values, the region R is divided into M juxtaposed right rectangular prisms composing the interpretative model. Using this approach, the gravity anomaly caused by the density anomalies can easily be approximated as the sum of the contribution of each prism using the formulas of Nagy et al. (2000). Assume that every individual prism has a constant density contrast; thus, the relationship between the gravity anomaly and the density contrast of each prism is linear and can be expressed in matrix notation as d= Gp (1), where d is the theoretical data vector, p is the parameter vector containing the density contrast of each prism, and G is the Jacobian matrix of partial derivatives with respect to the parameters. Inverse problem The inverse problem has the ability to become a non-linear, ill-posed, and under-determined problem. Solving the inverse problem is a process that involves combining the modeled data with the experimentally measured boundary data in the least-squares analysis. Solving the equation system shown in Eq. (1) for the vector p is an ill-posed problem and thus requires additional constraints. These conditions can be imposed by means of a regularizing function.

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Following this method, we formulate the inverse problem as a search for the minimum of the goal function: ΩðpÞ ¼ ϕd ðpÞ þ λϕp ðpÞ

ð1Þ

where λ is a regularizing parameter, the function ϕ d (p ) is a measure of the data misfit, and ϕ p (p ) is a regularizing function. Finding the minimum of the goal function and determining the density contrast of each prism are the main objectives of the inverse problems (Uieda and Barbosa 2011). Figure 1 shows how the forward and inverse models are processed. Regularized minimal residual method The goal of the inverse problem is to recover the physical property distribution that gave rise to the data in the leastsquares sense. A penalty term is added to the minimization system to obtain the stabilizing solution because the inverse problem is ill-posed, under-determined, and non-linear within this procedure. Therefore, the objective function is given by Ω ¼ ∥y−Gðμa Þ∥2 þ λ∥μa −μa0 ∥2

ð2Þ

where λ is the regularization parameter that will affect the resolution qualities of the result. The primary estimation of the absorption coefficient is given as μa0 . One of the techniques for finding the minima of the objective function (Eq. 2) is to make the first order condition with respect to μa equal to zero (full-Newton technique), which results in the subsequent revised equation:  −1 Δμa ¼ J T J þ λI J T ðy−Gðμa ÞÞ

ð3Þ

where J represents the Jacobian matrix [ J ¼ ∂G∂μðμa Þ of dia mension M×N, with M representing the number of measurements and N the number of finite element nodes], Δμa represents the update of μa, and I represents the identity matrix. The minimization problem given in Eq. (2) could be solved by applying the least-squares QR (LSQR) technique. As LSQR algorithms usually require only matrix–vector computations (as opposed to matrix–matrix computations), they provide greater computational efficiency than the classic approaches that are often used to solve the ill-conditioned inverse problems (Paige and Saunders 1982). This approach is iterative, naturally, as it searches for regularization that results in a unique solution. This regularization plays an important role in determining the quality of the results. A regularization parameter is typically selected either through the empirical choice or prior experience of the user.

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Fig. 1 Figure explaining the 3D forward and inverse problems in a gravity field. F is the forward modeling operator, m is the model, d is the data, d is density

LSQR-type method

To match y with G(μa), one can Taylor expand G(ua) around ua0, which is a guess of the original μa, giving

To solve the liberalized version of the objective function, a prior conjugate gradient type LSQR algorithm was used. The principle contribution of the LSQR-type algorithm arrangement is the estimation of the optimal regularization parameter. This formula is achieved by applying the Lanczos bidiagonalization of J (Paige and Saunders 1982). As further illustration, the left and right Lanczos matrices and the bidiagonal matrix related to the Jacobian matrix (J) are shown below: U kþ1 ðβ 0 e1 Þ ¼ δ

ð4Þ

J V k ¼ U kþ1 Bk

ð5Þ

J T U kþ1 ¼ V k þ BT k þ akþ1 nkþ1 eT kþ1

ð6Þ

Here, B is a lower bidiagonal matrix, and U and V are left and right orthogonal Lanczos matrices, respectively. The unit vector of dimension k×1 is represented by ek (=1 at the kth row and 0 elsewhere). U k ¼ M xk; V k ¼ N xk

ð7Þ

where k represents the number of iterations the bidiagonalization has performed. δ is the data-model misfit (δ=y−G(μa)), and ui and νi refer to the left and right Lanczos vectors, respectively. The structures of U and V are given by U k ¼ ½u1 ; u2 ; ::::::; uk ; V k ¼ ½v1 ; v2 ; ……; vk  Bk represents the bidiagonal matrix, where a1 …ak are on the main diagonal and b1 …bk are the lower sub-diagonal of the matrix with a dimension of ((k+1) xk).

Gðμa Þ ¼ Gðμa0 Þ þ J Δμa þ Δμa


T

HΔμa þ :::

ð8Þ

where J and T refer to the Hessian and the Jacobian, evaluated at μa0 and Δμa =μa −μa0, respectively. Now, by linearizing the resulting equation and assuming δ=y−G(μa0), a new objective function (linearized inversion) can be obtained: Ω ¼ δ − J Δma 2 ð9Þ

Substituting Eqs. (4), (5), and (6) into Eq. (9) gives   δ−JΔμa ¼ U kþ1 β0 e1 −Bk xðk Þ ð10Þ

where Δμa =Vkx(k). Substitution of Eq. (10) into Eq. (9) results in Ω ¼ β0 e1 −Bk xðk Þ 2 ð11Þ

Therefore, by replacing the changes in Eq. (11), the updated equation is xðk Þ ¼ BT k Bk þ λI


−1

β 0 BT k e1

ð12Þ

where β0 is the L2-norm of the data-model misfit (δ). When x(k) is obtained, Δμa(k) can be calculated by applying Δμa(k) = Vkx(k). In this approach, Δμa(k) is achieved through O(N2) operations, whereas the traditional approach (Eq. 3) requires O(N3) operations. As mentioned earlier, the number of iterations (k) has a significant role in determining the reconstructed result (Prakash and Yalavarthy 2013).

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Estimation of optimal λ using a LSQR-type method Discovering the dimensionality reduction capability is one of the advantages of using the LSQR-type method to find the update (Δμa). In comparison with the traditional method using Eq. (3), the updated method is more computational; therefore, by considering the positive side of computational efficiency, the new way of estimating the optimal regularization parameter (λ) is suggested by applying the simple method-based optimization scheme. In Eq. (9), the objective function is minimized with respect to the regularization parameter λ. As a result, in the update Δμa ¼ V k xðk Þ

ð13Þ

where x(k) is a function of λ, as given in Eq. (12). Moreover, k (the number of iterations of the Lanczos bidiagonalization) is the main element and has a significant role in this entire optimization scheme. In fact, k plays an important role in the computation of the optimal regularization parameter, and in addition, it ascertains the size of the bidiagonal matrix (Bk having a dimension of (k+1)×k). By increasing the number of Lanczos iterations, estimation of the optimal regularization parameter increases. This means that “the higher the k, the more is the ill-posedness of the problem,” while, in contrast, “the lower λ will become more optimal (similar to L-curve)” (Prakash and

Yalavarthy 2013). The appropriate algorithm for assessing the optimal number of iterations is given in algorithm 2. When k=50 (the number of iterations), the value of λ will be O(10−6), which refers to a single accurate limit. It is worthwhile to note that λopt in both the minimal residual method (MRM) and LSQR-based methods is searched for and calculated at each Gauss-Newton iteration. For the first iteration, the bound λlim for the optimal λ estimation is 1,000. Usually, a reduction of iterations occurs while estimating the optimal regularization parameter. It has been shown that from a computational aspect, the LSQRbased method estimation of optimal regularization is more efficient than the MRM-based estimation of the regularization parameter because its performance is based on repeated computation of the update using a sparse bidiagonal matrix (Eq. 12). MRM is based on the computation of O×(P×N2) (where P is the number of inner iterations), while LSQR calculates O×(2×Q×k2), where, in this case, Q is the number of function evaluations used for evaluating the optimal regularization parameter. If k (number of iterations) is very small, the best method is LSQR. In addition, the LSQR-type method is the updated equivalent of the Tikhonov regularization scheme when there is a minimization problem. Algorithm 1. Algorithm for determining the optimal number of Lanczos Iterations (kopt) and optimal regularization parameter λopt (Prakash and Yalavarthy 2013)

Input: Lanczos Bidiagonal Matrix Bk; Vk(k = 1, 2, ..., 50); , λ

0;

J,

a,

λlim

Output: Optimal number of Lanczos iterations: kopt and optimal regularization parameter:

opt

Initialize e1 for k = 1,2, ..., 50 1. Estimate the optimal λ for the given k (λ optk) –Use simplex method to find λ optk in the range of [0 λlim], which minimizes Eq. (10) (k) (k) a =Vk x , where x is found using Eq. (13).

with 2. Compute 3. Make

k

a

with λ= λ optk via Eq. (13).

a= a+

4. Estimate

k

a

and compute G(

=||y- G(

k

k

a)

(modeled data).

2 a)||

end kopt= index of minimal value of

k

and λ opt= λ optkopt

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The Lanczos bidiagonalization was performed using the MATLAB-based regularization tools, which is open source (Hansen 2007).

Euler deconvolution The objective of the 3D Euler deconvolution technique is to provide a map showing the locations and corresponding depth estimations of geologic sources of magnetic or gravity anomalies (Saibi et al. 2006, 2012) in a passing two-dimensional grid (Reid et al. 1990). The calculation of the Euler source points relies on Euler’s homogeneous Eq. (14), resulting in clusters used to constrain the final geometry of the model. The 3D style of Euler’s equation geometry is outlined (Reid et al. 1990) as

x

∂g ∂g ∂g ∂g ∂g ∂g þ y þ z þ ng ¼ x0 þ y0 þ z0 þ nb ð14Þ ∂x ∂y ∂z ∂x ∂y ∂z

∂g ∂g where ∂g ∂x; ∂y; ∂z are the derivatives of the field in the x, y, and z directions, b is the regional background of the field, and n is the structural index (SI) value that must be chosen to be consistent with a prior date of the source geometry. By considering four or more neighboring observations at once (an operated window), the source location (x0, y0, and z0) and b are computed by solving a linear system of equations generated from Eq. (14). Then, by moving the operated window from one location to the subsequent location over the anomaly, multiple solutions for a similar source are obtained. If the structural index n is known, one can use the least-squares method to obtain the unknown parameters of x0 ; y0 and z0 . As the Euler deconvolution method is employed for calculating the form and depth of the anomaly using gravity and magnetic resources, the interpreter does not clearly know the values of the two factors the moving window size and the SI. Therefore, in some cases, one can utilize a rule of thumb, prior information about the geology, or personal experience for determining the mentioned values. In this section, two loops are incorporated into codes written in the MATLAB language, despite previous works. In the first loop, all values of the SI from the set of 0–3 with an increase rate of 0.5 are included. With regards to the second loop, all values of the moving window size are included, i.e., odd values of 3–19, with an increase rate of 2. All the possible depths are extracted as well. Plotting the histogram of all Z values shows that the accepted values will be of the highest frequency, depending on the depth of the model (Toushmalani and Hemati 2013).

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Description of the new 3D gravity inversion algorithm As has been recognized, all the geophysical articles and references divide a specific area into a set of prisms (Parker 1973). In a majority of the articles (perhaps in all the articles) the minimum and maximum depths are not clear. The area discussed here is selected from the Bear article, where the depths have been calculated via the Euler deconvolution method. A code has been prepared in MATLAB that receives the desired area and divides it into a set of prisms, meaning that the desired area Xmin to Xmax, Ymin to Ymax, and Zmin to Zmax is divided into a set of prisms with the same or different dimensions; the number and dimensions of the prisms can be changed as desired. The noticeable point is that the considered area can be divided into 100,000 prisms that, according to the ability of the MATLAB programming language, can be performed simply and in a short time. The linear operation in the g=Aρ formula is used so that the solution of this equation is equivalent to the Tikhonov and Levenberg-Marquardt methods, where determining the value of λ is the most fundamental problem. Therefore, in this paper, the researcher had applied a least-squares QR (LSQR)-type method to determine the optimal value of λ. The least-squares QR (LSQR)-type method using the Lanczos bidiagonalization is known to be computationally efficient in solving inverse problems. The same method is effectively deployed via an optimization procedure that uses a simple method to find the optimal regularization parameter. The computational complexity of the proposed method is at least five times lower compared to other traditional methods, making it an optimal technique. Upon achieving the optimal value of λ and solving the above equation, the amounts of the density contrast of all the prisms are obtained, which are used in the next code section to investigate the considered densities. For instance, after solving the equation according to the mentioned levels in the considered model, the amounts of the density contrast are obtained. However, the author looks for those prisms with 400 density contrasts after running MATLAB code and is then asked to specify the range of the cubes that are being studied. Therefore, at this level, with devoting a trivial percent of error, 395 (minimum) and 405 (maximum), the amount of the differences can be determined. The considered code investigates all the prisms as well as just those prisms where the considered contrasts are determined. Then, they are depicted first in a 3D diagram and then in different forms of 2D diagrams. Regarding the considered region, we seek prisms with a density discrepancy of −0.2 because the density of the considered region is between 2.3 and 2.1 g/cm3 (the density of salt). In this case, those prisms with this density contrast are depicted. The flow chart in Fig. 2 explains the different steps of the newly developed algorithm for 3D gravity inversion.

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Fig. 2 The main flow chart of a 3D inversion algorithm of gravity data

Application to synthetic data

Fig. 3 Representation of original anomalous structure for first simulation test. Two geometrical bodies of positive density contrast 0.4 g/cm3 appear at different depths

In this section, results from the forward and inversion techniques as applied to synthetic data are presented. The synthetic model is composed of two buried anomalous bodies that are characterized by a density contrast of 0.4 g/cm3 with respect to the surrounding rocks and are more or less aligned along the xaxis. Figure 3 gives a view of this structure. The characteristics of the two anomalous bodies are as follows: body 1 with a mass of 264×1011 kg and a depth to its center Z1 =−134 m and body 2 with a mass of 360×1011 kg and a depth to its center Z2 =−360 m. First, we applied the 3D Euler deconvolution method to the synthetic model data. Most of the Euler solutions occurred in

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Fig. 4 All 3D and 2D results of the inversion of synthetic data (green cubes represent prisms with a density contrast of 0.4 g/cm3)

the vicinity of models with depths ranging between 0 and 500 m. Then, the model was divided into 960 prisms with dimensions of 200×200×50 m in the x, y, and z directions, respectively. Using the formula of Nagy et al. (2000), a coefficient matrix (prism gravity effect) was calculated, resulting in a

420×960 array matrix (being composed of 420 Bouguer anomaly measuring points and 960 prisms). Using the developed algorithm, all prisms with density contrasts of 0.4 g/cm3 were obtained and are shown in Fig. 4. It is noticeable that if the intended model is a combination of two or more objects with different density contrasts, we can do the search as many

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Fig. 5 a Elevation map from shuttle radar topographic mission (SRTM) data showing the location of the study area in Iran. b Geologic map of the study area

times as we wish using this method and depict the results each time in separate forms. Considering the intended model and the results achieved, this method can forecast the intended model with an acceptable rate; such a forecast includes the real depth and form of the anomaly source.

Application to real gravity data The case study is the Bandar Charak (also known as Dehnow) region, located in southern Iran (Fig. 5a). Geologically, Dehnow is a part of the Fars sedimentary basin in southeast Iran. Salt outcrops can be recognized at two points in the Dehnow anticline. The Khamy formation and the Bangestan group are the oldest geological structures in the area that have outcrops. Dominant structures trend NW–SE. The Dehnow anticline is located between the Hendurabi and Razak faults.

Fig. 6 Bouguer gravity anomaly map of the study area

These faults are nearly perpendicular to the Dehnow anticline. A low gravity anomaly is located in the southeast part of the anticline in the salt outcrop. A basic study of the geology of the area, a detailed investigation of the structural features (such as faults associated with the Dehnow anticline) and the application of geophysical techniques and other exploration methods are necessary to investigate the subsurface extension of this anticline and to identify the salt plug intrusion into the anticline. Gravity anomalies are the result of the interference between geological sources with different shapes, densities, and depths. Linear anomalies in geophysical maps, which may correspond to buried faults, contacts, and other tectonic and geological features, are particularly interesting for geologists. Based on recent density measurements from the study area, the mean density is approximately 2.3 g/cm3. The objective is to detect the salt dome in this area with a mean density of 2.1 g/cm3, which corresponds to a density contrast

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Fig. 7 All 3D and 2D results of the inversion of the case study area (green cubes represent prisms with a density contrast of −0.2 g/cm3)

of −0.2 g/cm3. Figure 5b shows the geologic map of the Bandar Charak area. Figure 6 shows the gravity anomaly map of the Bandar Charak area in units of milligalileos. For the real case study, the minimum and maximum depths were calculated between zero and 3,000 m using the Euler

deconvolution method. Afterwards, the study area was divided into 36,570 prisms with dimensions of 500×500×500 m in the x, y, and z directions, respectively. Using the formula of Nagy et al. (2000), a coefficients matrix was calculated, resulting in an 875×36570 matrix

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(875 gravity points and 36,570 prisms). Using the developed 3D gravity inversion algorithm, all prisms with density contrasts of −0.2 g/cm3 were detected and are shown in Fig. 7. In fact, prisms with such a density contrast are indicative of the figure of the salt dome situated within the study area, making these results compatible with the research findings from geological (Bosak et al. 1998) and geophysical investigations (Esmaeil Zadeh et al. 2010) in the study area.

Discussions and conclusions The new 3D gravity inversion method described above has several advantages over existing forward and inverse methods: 1. First, the method does not require transformation of data into the wave number domain and hence avoids the difficulties associated with Fourier transformation. 2. Second, the technique does not require data to be gridded. Randomly distributed observations will result in stable solutions. 3. The dimensions of the source volume, including minimum and maximum depth, are specified to match geological and physical constraints (with Euler deconvolution). 4. The new method can invert data sets with several thousand observations and up to a thousand unknown block densities in less than an hour. 5. In this study, we presented a new fast algorithm for 3D inversion of gravity data. The main characteristic of the study was implementation of the Lanczos bidiagonalization method, in which the forward operator matrix is replaced by a bidiagonal matrix with remarkably lower dimensions. Subsequently, the speed of the algorithm increased using this replacement. A comparison of the obtained results by the proposed method had close similarity to the solution of the original problem. 6. The proposed method provides very good geometry for the top of the models. 7. One of the abilities of this software is that it is not restricted to searching for a specific density contrast, but rather, the complete spectrum of the considered area density contrast is determined, and with the aid of the designed software, each specific density contrast is investigated. 8. Finally, a 3D model of the gravity source of the anomaly can be generated based on the density distribution. The limitations of the new method are summarized as follows: 1. The disadvantages of this gravity inversion approach come from the nature of the gravity field. First, the

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characteristic nonuniqueness of the solution requires the inclusion of constraints. 2. Any noise in the data is reflected directly in the inverse solution regardless of the source of the noise. 3. Prism cell size is reflected directly in the inverse solution, and finding an appropriate cell size is necessary. In conclusion, a new method for the 3D inversion of gravity data that uses a Lanczos bidiagonalization method was presented. Solving the full set of equations is very time consuming, and utilizing a good procedure to solve it approximately can cause a significant reduction in time. The Lanczos bidiagonalization technique can easily be an excellent algorithm to solve a Tikhonov cost function, significantly reducing the time required by substituting the forward operator matrix with a matrix of lower dimension. The method used to decrease the instability and to guarantee the uniqueness of the solution is to combine geological and geophysical constraints (density values, Euler deconvolution) into the inversion modeling. Synthetic and field data tests show that our method is able to recover an anomaly source with different density contrasts. Moreover, our method is able to handle irregularly sampled data. Our 3D model shows the shape of the salt dome in the study area.

References Abedi M, Gholami A, Norouzi GH, Fathianpour N (2013) Fast inversion of magnetic data using Lanczos bidiagonalization method. J Appl Geophys 90:126–137 Asli M, Marcotte D, Chouteau M (2000) Direct inversion of gravity data by cokriging. In: Kleingeld WJ, Krige DG (eds) Proceedings of the 6th International Geostatistics Congress—Geostat 2000. Geostatistical Association of South Africa, Marshalltown, pp 64–73 Barbosa VCF, Silva JBC (1994) Generalized compact gravity inversion. Geophysics 59(1):57–68. doi:10.1190/1.1443534 Barbosa VCF, Silva JBC, Medeiros WE (1997) Gravity inversion of basement relief using approximate equality constraints on depths. Geophysics 62(6):1745–1757 Bosak P, Jaros J, Spudil J, Sulovosky P, Vaclavek V (1998) Salt plugs in the eastern Zagros, Iran: results of regional geological reconnaissance. Ceolines 7:178 Bosch M, Meza R, Jiménez R, Hönig A (2006) Joint gravity and magnetic inversion in 3D using Monte Carlo methods. Geophysics 71(4):G153–G156. doi:10.1190/1.2209952 Bott MHP (1959) The use of electronic digital computers for the evaluation of gravimetric terrain corrections. Geophys Prospect 7(1):45–54 Camacho AG, Montesinos FG, Vieira R (2000) Gravity inversion by means of growing bodies. Geophysics 65(1):95–101 Camacho AG, Montesinos FG, Vieira R (2002) A 3-D gravity inversion tool based on exploration of model possibilities. Comput & Geosciences 28(2):191–204 Camacho AG, Fernández J, Gottsmann J (2011) The 3-D gravity inversion package GROWTH2.0 and its application to Tenerife Island, Spain. Comput Geosci 37(4):621–633

Author's personal copy 4980 Chasseriau P, Chouteau M (2003) 3D gravity inversion using a model of parameter covariance. J Appl Geophys 52:59–74. doi:10.1016/ S0926-9851(02)00240-9 Cheng D, Li Y, Larner K (2003) Inversion of gravity data for base salt: 73rd Annual International Meeting, SEG. Expanded Abstracts 588– 591 Danes ZF (1960) On a successive approximation method for interpreting gravity anomalies. Geophysics 25(6):1215–1228 Dias F, Barbosa V, Silva J (2009) 3D gravity inversion through an adaptive-learning procedure. Geophysics 74(3):I9–I21. doi:10. 1190/1.3092775 Eidsvik J, Avseth P, More H, Mukerji T, Mavko G (2004) Stochastic reservoir characterization using prestack seismic data. Geophysics 69(4):978–993. doi:10.1190/1.1778241 Esmaeil Zadeh A, Doulati Ardejani F, Ziaii M, Mohammado Khorasani M (2010) Investigation of salt plugs intrusion into Dehnow anticline using image processing and geophysical magnetotelluric methods. Russ J Earth Sci 11:1–9. doi:10.2205/2009ES000375 Franklin JN (1970) Well-posed stochastic extensions of ill-posed linear problems. J Inst Math Appl 31:682–716. doi:10.1016/0022247X(70)90017-X Fuentes M (2001) A high frequency kriging approach for non-stationary environmental processes. Environmetrics 12:469–483. doi:10.1002/ (ISSN)1099-095X Gallardo-Delgado L, Perez-Flores MA, Gómez-Treviño E (2003) A versatile algorithm for joint 3D inversion of gravity and magnetic data. Geophysics 68(3):949–959. doi:10.1190/1.1581067 Giroux B, Gloaguen E, Chouteau M (2007) Bh_tomo-a MATLAB borehole georadar 2D tomography package. Comput Geosci 33:126– 137. doi:10.1016/j.cageo.2006.05.014 Gloaguen E, Marcotte D, Chouteau M, Perroud H (2005) Borehole radar velocity inversion using cokriging and cosimulation. J Appl Geophys 57:242–259. doi:10.1016/j.jappgeo.2005.01.001 Green WR (1975) Inversion of gravity profiles by use of a Backus-Gilbert approach. Geophysics 40(5):763–772. doi:10.1190/1.1440566 Guillen A, Menichetti V (1984) Gravity and magnetic inversion with minimization of a specific functional. Geophysics 49(8):1354–1360 Haas A, Dubrule O (1994) Geostatistical inversion: a sequential method of stochastic reservoir modeling constrained by seismic data. First Break 12:561–569 Hansen PC (2007) Regularization tools version 4.0 for MATLAB 7.3. Numer Algoritm 46(2):189–194 Hansen TM, Journel A, Tarantola A, Mosegaard K (2006) Linear inverse Gaussian theory and geostatistics. Geophysics 71(6):R101–R111. doi:10.1190/1.2345195 Higdon D, Swall J, Kern J (1999) Non-stationary spatial modeling. Bayesian Stat 6:761–768 Krahenbuhl RA, Li Y (2002) Gravity inversion using a binary formulation: 72nd Annual International Meeting, SEG. Expanded Abstracts 755–758 Krahenbuhl RA, Li Y (2004) Hybrid optimization for a binary inverse problem, SEG International Exposition and 74th Annual Meeting, SEG. Expanded Abstracts 782–785 Last BJ, Kubik K (1983) Compact gravity inversion. Geophysics 48(6): 713–721. doi:10.1190/1.1441501 Lelièvre PG, Oldenburg DW (2009) A comprehensive study of including structural orientation information in geophysical inversions. Geophys J Int 178(2):623–637. doi:10.1111/gji. 2009.178.issue-2 Li Y (2001) 3-D inversion of gravity gradiometer data. SEG Technical Program Expanded Abstracts 1470–1473. doi: 10.1190/1.1816383 Li Y, Oldenburg DW (1998) 3-D inversion of gravity data. Geophysics 63(1):109–119. doi:10.1190/1.1444302 Li Y, Oldenburg DW (2000) Incorporating geological dip information into geophysical inversions. Geophysics 65(1):148–157. doi:10. 1190/1.1444705

Arab J Geosci (2015) 8:4969–4981 Li Y, Oldenburg DW (2003) Fast inversion of large-scale magnetic data using wavelet transforms and a logarithmic barrier method. Geophys J Int 152(2):251–265. doi:10.1046/j.1365246X.2003.01766.x Menke W (1989) Geophysical data analysis: Discrete inverse theory, Academic (1st Edition), 289 p Moraes RAV, Hansen RO (2001) Constrained inversion of gravity fields for complex 3D structures. Geophysics 66(2):501–510 Mosegaard K, Tarantola A (1995) Monte Carlo sampling of solutions to inverse problems. J Geophys Res 100(B7):12431–12447. doi:10. 1029/94JB03097 Nabighian MN, Grauch VJS, Hansen RO, LaFehr TR, Li Y, Peirce JW, Phillips JD, Ruder ME (2005) The historical development of the magnetic method in exploration. Geophysics 70(6): 33ND–61ND Nagy D, Papp G, Benedek J (2000) The gravitational potential and its derivatives for the prism. J Geod 74:552–560 Oldenburg DW (1974) The inversion and interpretation of gravity anomalies. Geophysics 39(4):526–536 Paciorek C, Schervish M (2006) Spatial modelling using a new class of nonstationary covariance functions. Environmetrics 17:483–506. doi:10.1002/(ISSN)1099-095X Paige CC, Saunders MA (1982) LSQR: an algorithm for sparse linear equations and sparse least squares. ACM Trans Math Softw 8:43–71 Parker RL (1973) The rapid calculation of potential anomalies. Geophys J R Astron Soc 31(4):447–455 Pedersen LB (1977) Interpretation of potential field data—a generalized inverse approach. Geophys Prospect 25(2):199–230 Pilkington M (1997) 3-D magnetic imaging using conjugate gradients. Geophysics 62(4):1132–1142. doi:10.1190/1.1444214 Portniaguine O, Zhdanov MS (2002) 3-D magnetic inversion with data compression and image focusing. Geophysics 67:1532–1541. doi: 10.1190/1.1512749 Prakash J, Yalavarthy PK (2013) A LSQR-type method provides a computationally efficient automated optimal choice of regularization parameter in diffuse optical tomography. Med Phys 40(3): 033101. doi:10.1118/1.4792459 Reid AB, Allsop JM, Granser H, Millet AJ, Somerton IW (1990) Magnetic interpretation in three dimensions using Euler deconvolution. Geophysics 55:80–91 René RM (1986) Gravity inversion using open, reject, and “shape-ofanomaly” fill criteria. Geophysics 51(4):988–994 Saibi H, Nishijima J, Aboud E, Ehara S (2006) Euler deconvolution of gravity data in geothermal reconnaissance; the Obama geothermal area, Japan. J Explor Geophys Japan (Butsuri-Tansa) 59(3):275–282 Saibi H, Aboud E, Ehara S (2012) Analysis and interpretation of gravity data from the Aluto-Langano geothermal field of Ethiopia. Acta Geophysica 60(2):318–336 Shamsipour P, Marcotte D, Chouteau M, Keating P (2010) 3D stochastic inversion of gravity data using cokriging and cosimulation. Geophysics 75(1):I1–I10. doi:10.1190/1.3295745 Shamsipour P, Chouteau M, Marcotte D (2011) 3D stochastic inversion of magnetic data. J Appl Geophys 73:336–347. doi:10.1016/j.jappgeo. 2011.02.005 Shamsipour P, Marcotte D, Chouteau M (2012) 3D stochastic joint inversion of gravity and magnetic data. J Appl Geophys 79:27–37. doi:10.1016/j.jappgeo.2011.12.012 Shamsipour P, Marcotte D, Chouteau M, Rivest M, Bouchedda A (2013) 3D stochastic gravity inversion using nonstationary covariances. Geophysics 78(2):G15–G24 Silva Dias FJS, Barbosa VCF, Silva JBC (2011) Adaptive learning 3D gravity inversion for salt-body imaging. Geophysics 76(3):I49–I57. doi:10.1190/1.3555078 Tarantola A (1988) The inverse problem theory: methods for data fitting and model parameter estimation. Elsevier, Amsterdam, 613 p

Author's personal copy Arab J Geosci (2015) 8:4969–4981 Tarantola A (2005) Inverse problem theory and methods for model parameter estimation: Society for Industrial and Applied Mathematics, 342 p Torres-Verdin C, Victoria M, Merletti G, Pendrel J (1999) Trace-based and geostatistical inversion of 3-D seismic data for thin-sand delineation; an application in San Jorge Basin, Argentina. Lead Edge 18: 1070–1077. doi:10.1190/1.1438434 Toushmalani R, Hemati M (2013) Euler deconvolution of 3D gravity data interpretation: new approach. J Appl Sci Agric 8(5):696–700

4981 Uieda L, Barbosa VCF (2011) 3D gravity gradient inversion by planting density anomalies. In 73rd EAGE Conference & Exhibition, Vienna, Austria, 23–26 May 2011 Zhang J, Wang C, Shi Y, Cai Y, Chi W, Dreger D, Cheng W, Yuan Y (2004) Three-dimensional crustal structure in central Taiwan from gravity inversion with a parallel genetic algorithm. Geophysics 69(4):917–924. doi:10.1190/1.1778235 Zhdanov MS, Ellis RG, Mukherjee S (2004) Three-dimensional regularized focusing inversion of gravity gradient tensor component data. Geophysics 69(4):925–937. doi:10.1190/1.1778236