euler deconvolution

Local phase filters such as vertical tilt derivative (TDR) and horizontal tilt derivative (TDX) are extensively used to interpret magnetic data. We use two combinations of these filters, namely TDR - TDX and TDR + TDX, to design a constraining mask that guides the Euler deconvolution moving data window. The TDR - TDX filter produces sharp peaks over the centers of the sources, while the TDR + TDX filter generates plateaus over them. Motivated by previous approaches that make use of Laplacian filter or analytic signal to constrain the Euler deconvolution window, we compute the solutions for windows centered at points that (1) have positive values of TDR - TDX and (2) are contained in the plateaus of TDR+ TDX. The use of both criteria improves the selection of source-related points while reducing the number of spurious ones. Our method is tested on synthetic anomalies due to dike-like sources, and also, on field data from an area in southeast of Iran. The experiments show that the use of a constraining mask based on combined tilt filters produces Euler solutions that are more contiguous and less sensitive to noise than the traditional methods.

The magnetic data collected from airborne surveys need to be interpreted after processing. The most important information gathered from the interpretation stage, is the depth and horizontal position of anomalies. Various methods are developed for gathering these data. One of these methods is Euler deconvolution that is based on Euler’s homogenous equation. Euler's method is one of the fastest semi-automatic methods for determining the depth of buried magnetic and gravity anomalies. Its results are highly dependent on the structural index, Euler window size, and depth calculation error. This method identifies the depth and trend of changes in the depth of anomalies very well. The geological information of the study area is important for the application of this method. In this method, the potential field and its first-order derivatives are used in different directions to determine the position and depth of the potential field source. In this paper, by using this method, the depth, and boundaries of anomalies in the Lake Blatchford area of Canada are investigated. The obtained results indicate that most anomalies in this region have shallow to medium depth.

Airborne magnetic and electromagnetic methods are among the most efficient geophysical techniques for the detection of buried anomalies. There are several methods that can be used to estimate the depths of the buried anomalies. In general, modeling methods can be used not only to estimate the depths of the buried anomalies, but also, to determine physical and other geometric factors of the anomalies such as lateral extension, thickness, dip and so on. In this research, magnetic lineaments have been determined using the airborne magnetic data, acquired in a part of Bazman area with an area of 24 square kilometers located in about 125 kilometers northwest of the city of Iranshahr. By applying filters such as reduction to the pole, first horizontal derivatives, analytical signal, tilt angle and upward continuation filters. For processing and interpretation of the airborne magnetic data, the Oasis Montaj module of Geosoft software package has been used. The processing, display and interpretation of the airborne electromagnetic data have been made Conductivity Depth Imaging (CDI) using EM Flow and Profile Analyst software packages of Encom Company. Furthermore, the depths of the lineaments in this area have been estimated using Euler deconvolution method. Then, the obtained results have been compared with the results of airborne electromagnetic investigations for the frequencies of 900, 7200 and 56000 Hz using horizontal and vertical coplanar coils. Also, the obtained findings from the airborne magnetic and electromagnetic methods have been validated by the geological information of the area. The airborne magnetic and electromagnetic data of the area have been acquired using airborne magnetometer and DIGHEM5 electromagnetic instruments, respectively. The airborne magnetic and electromagnetic surveys over the study area have been made by Geological Survey of Iran (GSI) in 2005. As a result of this study, 22 magnetic lineaments in the area have been identified in which 4 lineaments coincide on the main faults of the area as the validation results indicate. In this regard, the main faults can be observed on the obtained magnetic maps in which different filters have been applied, however, the tilt angle magnetic map indicates the main faults of the area more clearly. This implies the better performance of the tilt angle filter over the other filters in displaying magnetic lineaments. Totally, 22 magnetic lineaments have been determined on the magnetic maps. By the results of this study, we can conclude that the main faults of the area have an approximate trend of northeast-southwest. Some of these faults, which have been determined from the airborne magnetic investigations of the area, cannot be determined from geological studies of the area as they have been overlain by the Quaternary sediments. The different performances of these main faults on the lithological variations and tectonic activities of the area have been clearly evident by the result of this study. The main faults of the area have also played a vital role on the formation of folds and fractures, and occurrence of weak earthquakes. The approximate depths of the lineaments, which have been estimated by applying the Euler deconvolution method on the acquired magnetic data are around 100-200 meters.

Khak- Sorkh iron deposit lies about 42 km west of the Yazd city. The area is located in the central part of the Urumieh-Dokhtar magmatic belt and is covered by upper Triassic-lower Jurassic sedimentary units and Oligo-Miocene intrusions. Intrusion of Oligo-Miocene granitoids into Cretaceous limestones resulted in iron skarn mineralization. Magnetite is the main iron ore mineral; minor copper mineralization also occurs locally as malachite staining at surface. Two main structural trends can be distinguished: NW-SE which follows the main trend of the Urumieh-Dokhtar belt, and NE-SW trend that transects the former one and dissects magnetite bodies in some places.
In this study Euler deconvolution and analytical signal methods have been used for interpretation of geophysical data. Depth of magnetic anomalies determined by Euler deconvolution clarified that magnetic bodies are located at shallow depths in the NW of the area. The trends of the detected anomalies correlate very well with the general trend of the Urumieh-Dokhtar belt. This can be used as an exploration tool for magnetic anomalies in the area and likely in other parts of the Central Urumieh-Dokhtar belt. Based on magnetic anomalies 15 diamond bore holes were drilled at the NW part of the region which cut the ore bodies at predicted levels with average ore assay of 39 percent total iron.

Summary: Polour basalts are located 75 km away from Tehran. It is situated in northeast of Tehran and south of Damavand volcano. The basalt emission point has not been distinguished yet, whereas its occurrence is a debatable topic among geologists. The procedure of formation, geological setting and the way of ascending have been always as controversial issues. Petrologically, Polour basalts are basalt–basalt trachyte rocks. Similar units are found at the east of Damavand volcano and as expected, the composition, age and feeder source are different from Damavand volcano. Polour basalts occurred as arc-shaped bodies at shallow depth in central part of study region, where they mostly consist of old terraces and alluvial fans. Because of having considerable amount of magnetic susceptibility, magnetometric survey might delineate the location of such geological units. Various depth estimation techniques have been employed to determine the depth of causative sources associated with the basaltic units in the region of interest. Introduction: Several well-known techniques have been developed to estimate the depth of causative sources in potential field studies. Among them, AN-EUL, Euler deconvolution and analysis of power spectrum methods are the most efficient ones, which provide valuable pieces of information about the geometry of the prospect geological sources. Methodology and Approaches: AN-EUL technique, as a combination of the analytic signal and the Euler deconvolution methods, is an automatic algorithm for simultaneous estimation of depth, location and geometry of the subsurface sources in the potential field studies. The derivation of the main equations of this technique is based on the substitution of the derivatives of the Euler homogeneous equation into the analytic signal of the potential field data. Location of sources can be approximately estimated from the position of the maximum value of the analytic signal amplitude, and subsequently, the formulae of depth and structural index (SI) estimation are calculated at this point. An important advantage of the AN-EUL method is that it is not restricted only to idealized sources (i.e. having integer structural index). Its wider applicability means that the SI can be a fractional number that indeed describes sources with various arbitrary shapes. Analysis of power spectrum is also one of the methods used widely to estimate the depth of geological structures. The sources of magnetic anomalies within a region are assumed to average out so that spectral properties of an ensemble of sources are equal to the average of all causative sources responsible for potential field anomalies. This approach is advantageous since it is (1) statistically oriented, (2) averaging source depths over a region containing complex patterns of anomalies, (3) less affected by interference effects due to overlapping anomalies and high-wavenumber noise than other methods because it is based entirely on analyzing the wavelengths of the anomalies, (4) independent of the directional attributes of the magnetization of the sources and the geomagnetic field, and finally (5) it can be used to study a wide range of depths by varying the window involved in the data analysis. Results and Conclusions: Employing the above-mentioned methods for estimation of the depth and geometry of subsurface anomaly to the collected ground-based magnetometric data leads to obtain valuable information about the subsurface anomaly. Thus, in this study, the depths and structural indices of the subsurface anomalies have accurately been estimated that have been strongly in good agreement with the geological information of the study area. The maximum depth on both arcs is about 95 m below the surface topography. In addition, the 3D model from the area reveals the basalt root in both bodies amazingly is from separated sources. Due to the short length of profiles, that causes to have only shallow depths of identified bodies, the trend of creeping magma from the chamber is unknown. Subsurface anomalies and their depths indicate that the Polour basalts have deep roots and have not flown on the ground surface. By proving the existence of the anomalies as signatures of basalts, the depths and structural indices of the anomalies have accurately been estimated.

Depth estimation of geological structures is one of the most important objectives in geophysical studies. Euler deconvolution (standard Euler) is a well-known method in the depth estimation. Based on standard Euler, various methods are introduced to reduce error of the depth estimation. In this study we have used a new method called Euler RDAS. This method is based on the standard Euler. In this method derivatives of analytic signal and first vertical are used in Euler equation. Applying derivatives of analytic signal for depth estimation is better than the analytic signal. There is no problem of choosing structural index in this method which increases accuracy in the depth estimation. To examine the performance of this method, depths of several synthetic models are estimated and their results are compared to that of the standard Euler. In all models, results of the RDAS Euler shows fewer errors in high depth model in comparison to that of the standard Euler. This method was tested on synthetic data with noise. RDAS Euler sensitive to noise due to usage of high-degree derivatives is less than the standard Euler. Study shows that if the noise in the data is reduced by methods such as filter upward, it can provide an appropriate estimate of the depth. In this study, the depth of gravity anomalies caused by the masses of hematite located in Kerman, has estimated using standard Euler and RDAS Euler. Upward continued by 3 m has been used for reducing noise in this data. There are 3 possible hematite masses in residual map of the study area. The minimum of high depth anomalies indicates masses of hematite, which is calculated using RDAS Euler and was about 5 meters and a maximum of high depth is about 20 meters. The minimum depth obtained using standard Euler for this anomaly is about 5 meters and the maximum depth is about 40 meters. Responses of RDAS Euler is more compatible that of standard Euler with the boundary anomalies and has smaller vertical interval which can be a criterion for more precise solutions of the RDAS Euler. For further examination, the gravity data of hematite mine is used with inverse modeling of Camacho method. Minimum and maximum upper depths obtained for these anomalies are 5 to 35 meters, respectively. In addition, the modeling results is compared with the results of depth estimation of Euler. To this, 10 points in the anomalies area are pointed and the calculated depth of these points using standard Euler, RDAS Euler and modeling are shown. Root mean square error (RMS) between Euler’s and modeling results is calculated. In comparison of the results in different methods, which are not standard, results of two methods that have the lowest RMS error is considered as the selection criteria. RMS for standard Euler with MATLAB code, Geosoft, and RDAS Euler are equal to 11.43, 8.5, and 2.66 respectively. The results of two methods among these three methods which are used to estimate the depth of hematite masses are close hence they can be more reliable results. Therefore, due to fewer errors of RMS of RDAS Euler and modelling results are more accurate than that of the standard Euler.

The components of Gravity Gradient Tensor (GGT) is used for second-order derivatives of the gravitational potential field in the directions x, y and z in a Cartesian coordinate system. The third column of the gravity gradient tensor is Hilbert transform pairs of the first and the second columns. Many methods have been designed to estimate the depth, the horizontal position and the type of the sources from gravity gradient tensor components. Often, these methods are used derivatives of potential field data or their compounds in directions x, y, and z. Standard Euler deconvolution method is an approach in the interpretation of potential field data. It is able to locate the sources and to estimate the regional parameters with the assumption of the structural index. This approach is an automated method that has seen rapid development in recent years. The result of this method closely related to the precision of the assumed structural index parameter, and the accuracy is reduced in the presence of interference sources. Euler deconvolution of the directional analytic signal amplitudes is one of many methods to eliminate this problem. It is shown that the components of the gravity vector satisfy Euler's equation. Thus, it is proved that the amplitudes of directional analytic signal are homogenous and by putting in Euler's equation can estimate the location and the structural index of the gravity anomalies. In addition, two new equations were obtained from the combination of directional analytic signal amplitudes that is very effective in locating and estimating the structural index of gravity sources.
This paper was examined the application of Euler's equation of the directional analytic signal amplitudes to determine the location and the structural index of gravity anomaly sources. First, it is proved that each of directional analytic signal amplitudes in directions x, y, and z satisfy Euler's equation. Second, using the combination of directional analytic signal amplitudes derived two new equations that is more successful in determining the location (horizontal positions and depth) and source type (structural index) directly over the edges of gravity anomaly sources. The maxima of analytic signal amplitude in the z- direction place directly on the edge of the anomaly sources, but the maxima of analytic signal amplitude in the x- and y- directions deviate from the edges. That is why the simultaneous use of two or three-directional analytic signal amplitude can provide more accurate solutions.
The method described above was tested on the synthetic model in the presence of relatively high level Gaussian noise and interference sources. Finally, the method was applied to the Safoo manganese ore and obtained horizontal position, depth (~6 m) and structural index. MATLAB software was used to apply the above-mentioned methods.

The analytic signal method is a semiautomatic method for estimating the location of causative bodies in magnetic and gravity methods. The application of analytic signal for interpretation of two dimensional (2D) structures was introduced by Nabighian (1972). The analytic signal is defined as a complex function in which the real and imaginary parts are a pair of Hilbert transforms. In other words, the analytic signal is a combination of horizontal and vertical gradients of potential field. Analytic signal is a symmetric function with amplitude sensitive to parameters of the source. In case of 2D structures, the amplitude of the analytic signal is independent of the directional parameters such as inclination, declination and strike angle (Nabighian, 1972; Atchuta et al., 1981; Roest et al., 1992). The depth of 2D structures can be estimated using the width of the analytic signal or the ratio of the analytic signal to its higher derivatives (Hsu et al., 1996; Roest et al., 1992). Source’s parameters of a dyke such as width, dip, strike, magnetization and depth can be estimated by analytic signal method (Bastani & Pedersen, 2001). The nth-order enhanced analytic signal is defined as the analytic signal of the nth-order vertical derivative of the potential field. An automated method for estimating the depth, horizontal location and shape of 2D magnetic structures is the horizontal gradient of analytic signal method. This method is capable of interpreting low quality data because of using the first and second order derivatives of potential field in the main equations. The method of analytic signal estimates the horizontal location of the source by approximating the maximum amplitude of the signal; hence noise can affect the estimations. On the other hand, by using the horizontal gradient of analytic signal expressions, all of the source’s parameters could be approximated simultaneously. In this method, equations of the analytic signal, Euler enhanced analytic signal and horizontal gradient of analytic signal are combined to derive a linear equation. Using the first order analytic signal, horizontal gradient of analytic signal and linear inversion method, the depth and horizontal location of 2D magnetic bodies are obtained. The location estimation is independent of the shape of the causative bodies. The causative body’s geometry is estimated as a structural index by applying the least squares method. Data selection for solving the equations or width of windows is based on data quality. The optimum size is defined somehow to detect a signal specific anomaly and also variations of the anomaly in one window. In this study, in order to solve the equations of the horizontal gradient of analytic signal method, the data greater than twenty percent of maximum amplitude of the analytic signal were used. The analytic signal-Euler deconvolution combined method is an automated method to estimate depth and shape of the sources. This method is used to interpret 2D & 3D magnetic and gravity data. After substituting the appropriate derivatives of the Euler’s homogeneous equation in the equation of the analytic signal, major independent equations which are used to estimate the depth and shape of causative bodies, are derived. The horizontal location of causative bodies is estimated by Euler method or locating the maximum amplitude of the analytic signal. In this study, the accuracy and efficiency of each of the mentioned methods in interpretation of magnetic anomalies are evaluated. Methods were tested for different synthetic datasets provided by forward modeling. 2D magnetic models placed at different depths and random noise added for some models. Derivatives were calculated in frequency domain by using Fourier transform techniques. In this technique, bell-shapedness effect appears at the edges of the profiles. This effect could be corrected by linearly expanding the profiles. Upward continuation filter was applied on some synthetic data to decrease the noise level. In this paper, the applicability of the horizontal gradient of analytic signal method and the analytic signal-Euler combined method were tested. Both methods estimate the parameters of the causative bodies without any prior information. In both methods, there is not any explicit dependence on directional parameters (e.g. magnetization) in the main equations; hence, as the results show, estimations were not affected by remanent magnetization. The results also show accurate estimations of the horizontal gradient of analytic signal method for shape and horizontal location and efficient estimations of the analytic signal-Euler deconvolution combined method for depth.

Horizontal and vertical gradients of the potential fields are used routinely to enhance the edge of the magnetic and gravity sources; furthermore, they are used as useful tools in interpreting and processing of magnetic and gravity data. In general, the derivatives of the potential fields are divided into horizontal and vertical derivatives, and they have always been significant tools in interpreting and processing of potential data. The derivatives can be determined in two procedures, direct measuring when the data are recorded, and calculation using mathematical and numerical methods. Many interpreting methods, that estimate the depth, location and the shape of a potential source, are based on using the gradients of potential fields. For example, both analytic signals and Euler Deconvolution methods, that have been widely applied, basically use the potential field derivatives. In these methods, different kinds of first order derivatives or derivatives of other positive integer orders are commonly used. In the basic equations of these methods, it is possible to use the derivatives of fractional orders in place of derivatives of other positive integer orders. Derivatives are high pass filters. They intrinsically amplify any noise and shallow anomalies present in the data. Therefore, using high order derivatives would be less common. Instead of using high order derivatives, one should use fractional order derivatives of the field. Besides, negative order derivatives are applicable in these kinds of methods and equations, and they can be considered as an interesting property of negative order derivation that acts as a low pass filter. In addition, horizontal fractional derivatives can be used instead of reduction to the pole at low latitudes to eliminate the instability of the reduced data. In this paper, the methods of the field gradient calculation, their alternation, and the application of the fractional order derivatives in analytic signals and reduction to the pole were inquired. To study the effects of the derivatives of different orders, the method was applied to synthetic data generated by various magnetic models such as a thin dike, and a horizontal cylinder. In the next stage, to simulate the real cases, the data was contaminated by random noise. To produce the synthetic data, the forward modeling was used. Finally, the method was applied to an aeromagnetic data set acquired over an area in Sweden. According to the geological studies in this region, there exists a granite intrusive body with certain fractures in which Diabase veins have penetrated. The results show that the fractional order derivatives as well as negative order ones are useful in data processing, and they can be considered as the principle of some of interpreting methods. All of the processing steps in this paper have been performed by using the code that we have written in Matlab.

AN-EUL is a new automatic method for simultaneous estimation of depth, location and geometry of magnetic and gravity sources. The principle of this method is a combination of both the analytic signal and Euler Deconvolution methods. The derivation of the main equations of this method is based on the substitution of the appropriate derivatives of the Euler homogeneous equation into the expression of the analytic signal of the potential field. Location of source (Epi-centre) can be approximately estimated based on the position of the maximum value of the amplitude of the analytic signal, and the formulas of depth and structural index (SI) estimation is calculated at this point. This new method is applicable on data along profiles and/or girds. It is one of the basic characteristics of the analytic signal applied on the responses of the two dimensional magnetic sources, such as dike and infinitely long horizontal cylinders, that the shape of the signal amplitude and its location are independent of the magnetization direction. For these types of sources, the shape of the amplitude of the analytic signal is symmetrical, whereas for 3-dimensional sources, like spherical sources, the maximum value of the amplitude of the analytic signal is not always located directly over the body, and, for these sources, the shape of the amplitude of the analytic signal depends on the direction of magnetization and is asymmetric. Therefore, there will be some errors in determining the location of the magnetic source based upon the location of the maximum value of the amplitude of the analytic signal for these types of sources. An important advantage of the AN-EUL method is that it is not restricted only to idealized sources (i.e. having integer structural index). This wider applicability means that SI can be a fractional number that describes sources with arbitrary shapes. Because of the existence of high order derivatives in the AN-EUL method formula, this method is very sensitivity relative to noises and shallow sources; thus, the effects of noises and shallow sources can be reduced by applying an upward continuation filter. To study the resolution of the AN-EUL, the method has been applied on synthetic data generated from various magnetic models, including a thin dike, a magnetic sphere and a drum shape source. In the next stage, the simulation of real cases, the data were contaminated by random noise. For all of these models, with regard to the models parameters, the results have good accuracy. Finally, the method was applied to an aeromagnetic data set acquired over an area in Sweden to estimate the depth, location, and shape (structural index) of some of the anomalies. According to the geological studies in this region, there exists a granite intrusive body with certain fractures in which Diabase veins have penetrated. Results of this study show the nature of anomalies very well and give good estimations of the depth and shape of the magnetic sources causing these anomalies. The results agree well with the geological information found by other methods (e.g. MT, Gravity, field observations). All of the processing steps in this paper were performed by using codes wrote in Matlab.

AN-EUL is a new automatic method for the simultaneous approximation of depth, geometry and location of magnetic sources. The principle advantage of this method is its combining both the analytic signal and the Euler Deconvolution methods. It is by substituting the appropriate derivatives of Euler homogeneous equations for the expression of the analytic signal of the potential field in the main equations of this method that the advantage is gained. In this method, the determination of the source location is based on the position of the maximum value of the analytic signal amplitude. For 3D sources, the maximum value of the amplitude of analytic signals (AAS) is not always located directly over the body, and the shape of AAS is dependent on the magnetization direction. Therefore, there are some errors in the determination of location based on the maximum value of AAS for these types of sources. Consequently, the calculation of the depth and structural indices, also have errors when using the maximum value of AAS. By using the reduction to the pole and pseudo gravity transform of the magnetic anomaly, the approximation of the source location and, consequently, the calculation of the depth and structural indices of the source gain high accuracy. In this paper, in order to compare results, the AN-EUL method has been applied to magnetic data, reduced to the pole data and pseudo gravity data due to the magnetic sphere. The results show that the calculations have greater accuracy for reduced to the pole data and pseudo gravity data than for magnetic data. Finally, this method is applied to a series of real magnetic data. By using the reduction to the pole and pseudo gravity filters, the reduced to the pole and pseudo gravity anomalies are calculated. Subsequently, the AN-EUL method is applied to these data, and the results are compared with each other. All of the steps in this paper are performed by using a written code in MATLAB.