Summary: Chromite exploration is really important in mineral exploration. Gravity method is really important in chromite exploration. Edge detection methods are used to determine lenses of chromite. In this paper, we used the curvature gravity Gradient Tensor (CGGT) along with the tilt angle method to detect chromite lenses. Application of the methods on synthetic and real gravity data showed that the CGGT can determine the edges of chromite lenses better than the tilt angle method. Introduction: Chromite is a strategic mineral. Therefore, the exploration of chromite mineral reserves is the main mineral exploration priorities. Chromite has a marked density contrast with the host rock, so the gravity method can be applied for exploration of the chromite ore bodies. The boreholes locations are usually determined after finding the edges of the chromite lenses by edge detection of the gravity anomalies. There are various edge detection methods. Most of the edge enhancement techniques are interpreted qualitatively. The Tilt angle method is a traditional method that can detect edges of subsurface structures quantitatively. The value of Tilt angle is zero above edges of subsurface bodies. The curvature gravity Gradient Tensor (CGGT) was also used to interpret the geological structure quantitatively. The value of eigenvalues of CGGT are zero above edges of subsurface bodies. In this paper, we used CGGT for edge detection of chromite lenses. Methodology and Approaches: In order to obtain CGGT, at first, horizontal vector Gradients of gravity Gradient Tensors are computed from the vertical component of gravity data with a Fourier transform technique. Then the eigenvalues of CGGT are obtained. The large eigenvalue determines the edges of negative density bodies while the small eigenvalue only can be used to outline edges of positive density bodies. The chromite has positive density contrast with the host rock and produce positive gravity anomaly. Therefore, we choose the small eigenvalue to outline edges of the chromite lenses. Finally, the tilt angle is also applied to compare with the CGGT. Results and Conclusions: The robustness of the codes used for the edge enhancement is tested with gravity field anomaly map caused by four prisms of synthetic bodies. The results indicated that the proposed method can enhance the edges of the synthetic bodies with zero contour of the small eigenvalue of the CGGT. Then, the proposed method has been applied on the real gravity data from chromite deposits In Camaguey province, Cuba. The results showed that the zero contour of the small eigenvalue of the CGGT can outline the edges of synthetic bodies and chromite lenses better than the zero contour of the tilt angle method. Therefore, we can use the small eigenvalue of the CGGT to detect edges of chromite lenses precisely.

Edge detection of subsurface structures is an important objective in interpretation of magnetic data. In this paper, curvature Gradient Tensor (CGT) of magnetic data has been used along with tilt angle method to detect edges of subsurface structures. Application of these methods on synthetic and real gravity data has shown that the CGT of magnetic data, compared to the tilt angle method, can determine the edges of subsurface structures better. Introduction The main objective of the interpretation of magnetic data is to extract information about subsurface structures. Edge detection is an important means to image the edges of subsurface structures. Therefore, edge detection has traditionally been an important objective in the interpretation of magnetic data. There are various methods for edge detection. Tilt angle method is a traditional method that can detect edges of subsurface structures quantitatively. The value of Tilt angle is zero above the edges of subsurface bodies. The curvature gravity Gradient Tensor (CGGT) has also been used to interpret subsurface geological structures quantitatively. The eigenvalues of CGGT are zero above edges of subsurface bodies. In this paper, the CGT of magnetic data has been used for edge detection of subsurface magnetic bodies. The results of using the CGT of magnetic data have been compared with the results obtained from applying Tilt angle method on the data. Methodology and Approaches In order to obtain the CGT of magnetic data, at first, the magnetic data are reduced to pole (RTP). Then, horizontal vector Gradients of the Gradient Tensors are computed from the RTP data using a Fourier transform technique. Then, the eigenvalues of the CGT of magnetic data are obtained. The small eigenvalue can only be used to detect the edges of bodies with positive susceptibility contrast, and the large eigenvalue can only be used to determine the edges of bodies with negative susceptibility contrast. As an example, chromite ore has positive density contrast with the host rock and produce positive gravity anomaly. Finally, the tilt angle method is also applied to compare its results with those of the CGT of magnetic data. Results and Conclusions The robustness of the method used for the enhancement of edge detection is tested with a magnetic anomaly map caused by two prisms of synthetic bodies with positive and negative susceptibility contrast. The results have shown that the zero contour of the small eigenvalue of the CGT of magnetic data compared to the zero contour of the tilt angle method can better detect the edges of synthetic bodies with positive susceptibility contrast. Moreover, the zero contour of the large eigenvalue of the CGT of magnetic data compared to the zero contour of the tilt angle method can better detect the edges of synthetic bodies with negative susceptibility contrast. The Tilt angle method is also more sensitive to noise than the CGT of magnetic data. The CGT method has been applied to real magnetic data from Qahan porphyry copper deposit in Markazi Province, Iran. The results have indicated that the large eigenvalue of the CGT can determine the edges of porphyry deposit and the small eigenvalue can outline positive magnetic anomalies caused by propylitic alteration. However, the tilt angle method has not been capable of finding the edges of the porphyry deposit.

One of the methodologies employed in gravimetry exploration is eigenvector analysis of Gravity Gradient Tensor (GGT) which yields a solution including an estimation of a causative body’ s Center of Mass (COM), dimensionality and strike direction. The eigenvectors of GGT give very rewarding clues about COM and strike direction. Additionally, the relationships between its components provide a quantity (I), representative of a geologic body dimensions. Although this procedure directly measures derivative components of gravity vector, it is costly and demands modern gradiometers. This study intends to obtain GGT from an ordinary gravity field measurement (gz). This Tensor is called Computed GGT (CGGT). In this procedure, some information about a geologic mass COM, strike and rough geometry, just after an ordinary gravimetry survey, is gained. Because of derivative calculations, the impacts of noise existing in the main measured gravity field (gz) could be destructive in CGGT solutions. Accordingly, to adjust them, a “ moving twenty-five point averaging” method, and “ upward continuation” are applied. The methodology is tested on various complex isolated and binary models in noisy conditions. It is also tested on real geologic example from a salt dome, USA, and all the results are highly acceptable.

Magnetometry is one of the geophysical methods that has wide applications. In traditional magnetometry, one sensor so-called magnetometer is used for measurements. In order to obtain the Gradient in a specific direction, mathematical derivation can be used. Alternatively, A more accurate method of obtaining the Gradient is using two sensors simultaneously measuring the magnetic field. In this case, the Gradient is obtained in the direction where the two sensors are placed relative to each other. To obtain the Full Magnetic Tensor Gradient (FMTG) of the Earth’s magnetic field, although direction derivatives can be used, the more accurate method is to use four sensors that measure the Earth’s magnetic field simultaneously. In this way, the FMTG matrix can be obtained. So far, Superconducting Quantum Interference Device (SQID) sensors have been used the most in obtaining the FMTG. But these sensors are very expensive and have accurate performance in a small range of temperatures. By developing the Micro Electro Mechanical System (MEMS) sensor with its variety of applications, geophysicists are also becoming interested in using such sensors. In this study, one of the precise MEMS magnetometers has been selected. Although the MEMS sensors have low sensitivity compared to SQUID magnetometers, due to their availability, being cheap, and lightness, they are a suitable option for FMTG measurements. Although these sensors do not have the resolution of SQUID magnetometers and they cannot be used on small anomalies that have caused changes of less than 160 nT on the Earth’s magnetic field, these kinds of sensors are able to measure the magnitude of the magnetic field in three perpendicular directions and have an acceptable sensitivity. In this study, four sensors of this type have been set up in a cross arrangement. In order to solve the problem of the low sensitivity of these sensors compared to SQUID sensors, the distance between the sensors was chosen to be one meter so that the Gradient value would be larger enough. Then, a survey has been performed on an iron ore deposit near Jafar Khan village, Saqqz City, Kurdistan province. In order to validate elements of the FMGT matrix, they have been compared with the directional derivatives obtained from a sensor. This comparison shows that the elements Signal-to-noise ratio of elements Gxx, Gyx, Gzx, Gzy, and Gzz have increased around zero, 4.79, 10.80, 83.83, and 8.46, respectively, compared to their corresponding directional derivatives. This issue shows the ability of MEMS sensors to capture the full Tensor of the magnetic Gradient.

Summary Edge detection of causative bodies is important in the interpretation of potential field data. There are many methods that can be employed to detect and enhance the edges. In recent years, many filters have been presented based on potential field Gradient Tensor data. Because of using nine signal components, these methods have a very high accuracy compared to previous proposed methods in this regard. In this paper, normalized horizontal modulus (NHM) method, based on the potential field Gradient Tensor data, has been proposed. This method makes use of the modulus of potential field Gradient Tensor to normalize the horizontal modulus of potential field Gradient Tensor. The minimum value of NHM specifies the edge of magnetized structures. The proposed method with very high degree of precision delineates the edges of anomalies and does not show distraction. This new filter is tested on synthetic data and finally, it has been applied to the aeromagnetic data of the Varzaghan area, and as a result, the location of faults in the area has been determined with high accuracy. Introduction The potential-field Gradient Tensors are the second derivatives of potential-field data. Since the potential-field Gradient Tensor data contains nine signal components, their interpretation allows a high resolution and detailed investigation of geological structures. The methods based on the potential field Gradient Tensor (PGT) matrix use curvature or eigenvalue of the PGT matrix or directional methods of the PGT matrix. In the directional methods of the PGT matrix, in order to provide a more detailed map of the subsurface, Oruç and Keskinsezer (2008) have defined the directional tilt angles filters. Mikhailov et al. (2007) and Beiki (2010) have proposed the directional analytic signal to delineate the edges. However, this technique cannot display the edge of amplitude size of different anomalies simultaneously. Yuan and Yu (2015) have introduced second order directional analytic signal method and then, proposed a normalization method, which can display the large and small amplitude edges simultaneously. Yuan et al. (2016) have proposed horizontal directional theta (ED) method as an edge detection method based on the gravity Gradient Tensor that can weaken the above defects. It has higher resolution compared to the previous filters for delineating edges. However, for complex geological situations, the mentioned methods have some restrictions for edge detection. In this paper, to overcome these restrictions, NHM method has been proposed, and compared to other methods produces more detailed results. Methodology and Approaches The NHM is defined as follows: where HM is the horizontal modulus of the potential field Gradient Tensor and M is the modulus of potential field Gradient Tensor. The minimum value of NHM specifies the edges of magnetized structures. The NHM method beside tilt angle, total horizontal derivative of the tilt angle (THDR) and ED methods has been applied on synthetic magnetic data, and their results have compared. Moreover, the NHM is applied on aeromagnetic data from Varzeghan area. Results and Conclusions The results of applying the tilt angle, THDR, and NHM methods on synthetic magnetic data have shown that the THDR method has been succeeded in determining the position of the bodies, however, the edges of the deeper bodies have not been recognized clearly. The tilt angle method can recognize the edges of the shallow and deep bodies simultaneously but with low accuracy. The ED method possesses high precision and high accuracy in identifying the edges, but in its results, some distortions can be observed. The NHM method of the total intensity data can display the edges of the bodies more accurately while no distortion is seen in the results of the NHM method. Therefore, the NHM filter, compared to other filters, produces more detailed results. The proposed method has been applied on the aeromagnetic data from Varzeghan area, and as a result, the recognized edges of the geological structures are found to be precise and clear.

Accurate modeling of the sub-grid scales (SGS) is crucial in determining the accuracy of the large eddy simulations (LES) in turbulent flow analysis. In recent years, new branches of the sub-grid scales models called Gradient-based models were developed in computing the sub-grid scales stresses and heat fluxes and used in large eddy simulations. In this work, the modulated Gradient model (MGM) equations were implemented in the Open FOAM package, and pimple Foam solver was modified to improve the solution accuracy. The modulated Gradient model is based on the Taylor-series expansion of the sub-grid scales stress and employs the local equilibrium hypothesis to evaluate the sub-grid scales kinetic energy. To assess the accuracy of the modulated Gradient model as well as the improved pimple Foam solver, turbulent channel flow at a frictional Reynolds number of 395 was simulated via the Open FOAM package and results were compared with the direct numerical simulation (DNS) data as well as the numerical solution of the Smagorinsky, Dynamic Smagorinsky and Deardorff models. The results show that modulated Gradient model evaluates first and second order turbulence parameters with a high-level of accuracy.

Multilinear Discriminant Analysis (MDA) is a powerful dimension reduction method specifically formulated to deal with Tensor data. Precisely, the goal of MDA  is to find mode-specific projections that optimally separate Tensor data from different classes. However, to solve this task, standard MDA methods use alternating optimization heuristics involving the computation of a succession of Tensor-matrix products. Such approaches are most of the time difficult to solve and not natural, highligthing the difficulty to formulate this problem in fully Tensor form. In this paper, we propose to solve multilinear discriminant analysis (MDA) by using the concept of transform domain (TD) recently proposed in [15]. We show here that moving MDA to this specific transform domain make its resolution easier and more natural. More precisely, each frontal face of the transformed Tensor is processed independently to build a separate optimization sub-problems easier to solve. Next, the obtained solutions are converted into projective Tensors by inverse transform. By considering a large number of experiments, we show the effectiveness of our approach with respect to existing MDA methods.

With the appearance of the satellite altimetry in 1973, a new window was opened in the oceanography, marine sciences and Earth-related studies. Advances in the sensors technology and different satellite altimetry missions in the recent years led to a great evolution in geodesy and the gravity field modeling studies. Satellite altimetry provides a huge source of information for the geoid determination with high accuracy and spatial resolution. The information from this approach is a sufficient alternate for the marine gravity data in the high frequency modeling of the Earth’s gravity field in marine areas. Marine gravity observations always carry a high noise level due to the environmental effects. Moreover, it is not possible to model the high frequencies of the Earth’s gravity field in a global scale using these observations. The gravitational Gradient Tensor, as the second order spatial derivatives of the gravitational potential, provides more information than other measurements from the Earth’s gravity field such as the gravity anomaly. In this paper, a new approach is introduced for the determination of the gravitational Gradient Tensor at sea level based on the satellite altimetry and using two modeling techniques, namely radial base functions and harmonic splines. As a case study, the gravitational Gradient Tensor is determined in Persian Gulf based on the satellite altimetry data, and the results are presented. By the investigation of the results for the gravitational Gradient Tensor, it is concluded that modeling of the Earth’s gravity field using radial base functions leads to better results compared to the modeling based on the harmonic splines.

Gravity Gradient Tensor is a matrix containing the second order derivatives of the Earth’s gravity field, which has numerous applications in geodesy and geophysics. To date, much effort has been done for estimating gravity Gradient Tensor with reasonable accuracy. This quantity can be estimated via using various methods, and one of these methods is applying finite-difference method to gravity observations. Finite-difference method can estimate gravity Gradient Tensor directly by using the mathematical concept of Gradient, regardless of extra assumptions. This ability of finite-difference method, from theoretical point of view, provides the possibility of accurate estimation of gravity Gradient Tensor without considering additional assumptions to the problem. This study tends to introduce and evaluate Finite difference method for estimating the Gradient Tensor and present formulae for determining gravity Gradient Tensor from land-based gravity observations. In this paper, the proposed equations are numerically tested by means of using a global gravity model of the earth. Global gravity model of the earth (EGM 2008) is a geopotential model of the earth consisting of spherical harmonic coefficients up to degree 2190 and order 2159. There are numerous uses for these high degree potential coefficient models. One of these uses is modeling and estimating gravity Gradient Tensor.Finally, gravity Gradient Tensor is estimated by the proposed method for 6350 gravity stations located in Costal Fars region in a northern part of the Persian Gulf, between the latitudes from 26.5 N to 27.27 N and longitudes from 53.41 E to 55.58 E. The target area is about 10000 square kilometers. About 8500 square kilometers of the study area is located in moderate mountainous regions, and about 1500 square kilometers is located in flat coastal areas. The altitudinal distribution and spatial distribution of gravity in study area are shown in figure 1 and 2 respectively. Numerical experiments of this study demonstrate the ability of this method in gravity Gradient Tensor estimation with acceptable accuracy. For example, numerical experiments showed that the proposed method can estimate diagonal components of gravity Gradient Tensor (second order derivatives of the Earth’s gravity field in east, north and vertical directions) with the accuracy values of 12.46, 34.49 and 454.82 Eotvos respectively. The spatial distribution of the gravity Gradient Tensor components obtained from finite difference method in study area are shown in figure 3.Finally, according to the theoretical concepts discussed in this paper, It can be said because the finite difference method using from derivative and difference concepts directly for estimating gravity Gradient Tensor, it is expected that this method provide accurate estimation of gravity Gradient Tensor, As this is happen in the simulation conducted. However the accuracy of this method is very dependent on distances between sampling stations and by reducing distance between the stations, the accuracy of proposed method will be increased. The numerical results of this study also showed that the proposed method can provide accurately estimate of gravity Gradient Tensor components In some stations surrounded by suitable spatial distribution of gravitational observations.