جستجوی مقالات مرتبط با کلیدواژه « finite difference » در نشریات گروه « زمین شناسی »

Summary: The resistivity method is frequently used in fields of engineering geology and exploration of mineral resources. The simplicity of the equipment, the low cost of the survey in comparison with other methods, and the abundance of interpretation methods makes it as a popular geophysical method. There are many methods for inversion of resistivity data. Validation of inverted models is an important step in modeling. In general, validation of the resistivity results is performed by calculating the difference between observed and estimated values, but in this study, a validation technique based on data resolution matrix and model resolution matrix is proposed. The applied method for validation of the results has not been used so far, in this research work, resistivity and induced polarization data have been collected using dipole-dipole electrode array in Hamyj copper deposit located near the city of Birjand. The results of data resolution matrix and model resolution matrix have shown that generalized inversion is a suitable method for processing of resistivity data, because both data and model resolution matrix have been close to an identity matrix.

Introduction: After gathering the resistivity data, application of a suitable inversion method for finding an adequate subsurface model is very important. Visual and analytical methods are used for the interpretation of resistivity data over simple structures such as faults. However, these methods require a certain degree of symmetry and they are suitable only for simple geological conditions. Generalized inversion is one of the important modelling techniques to invert geophysical data. In current study, generalized inversion is used for inversion of resistivity data. Normally the validation of the resistivity results is performed by calculating the difference between observed and estimated values, i.e. error function, but data resolution matrix and model resolution matrix are suitable tools for validation of the results. This method for validation of the results has not been used so far. In this research work, unit covariance matrix has been used to identify the correctness of the each parameter. Moreover, data resolution matrix describes the accuracy level of the estimated values. The covariance of the model parameters depends on the covariance of the data and the way that the error is mapped from data to model parameters. This mapping is just dependent on the data kernel and the generalized inversion, which is independent of the data.

Methodology and Approaches: The proposed technique was tested on synthetic and real datasets. To explore the capability of the applied method more, 10 percent noise was also added to the synthetic data. The results of synthetic dataset showed the capability of the applied technique in the absence and presence of noise. For collecting the real resistivity data, an electrode spacing of 20 m has been used. Then inversion of the resistivity data acquired along a survey line was carried out using the generalized inversion method. Finite difference method was used for forward modelling, and also, perturbation approach was used for calculation of the Jacobin matrix in the inversion process. The results showed the presence o a fault in the study area. Furthermore, the results had a good correlation with the geological evidence from the study area. In this research, the code of the generalized inversion method has been written in MATLAB.

Results and Conclusions: In the present study, a new method for the inversion of resistivity data has been proposed. The proposed method, which is a, generalized inversion method, has been tested on synthetic and actual datasets. The results have shown that the generalized inversion method is a successful technique in the inversion of the resistivity data, because both data resolution matrix and model resolution matrix have been close to an identity matrix. The results, obtained from applying the unit covariance matrix, have shown that the variance of some data is not zero. In other words, the field datasets acquired from the southeast of the survey line, have less accuracy. Finally, we can conclude that these three matrixes for the validation of the model and finding the best model parameters are very useful.

Today, geophysical methods are one of the most feasible means for exploration, detection and localization of underground targets. One of the most used geophysical methods is ground penetration radar (GPR). In this method, based on the depth, shape and electromagnetic properties of the target and surrounding environment, electromagnetic waves with different central frequencies are propagated through the medium. In this paper, first, the theory of GPR, and then, forward modeling of GPR data using finite difference time domain (FDTD) method with consideration of convolutional perfectly matched layer (CPML) absorbing condition are presented. Finally, forward modeling of a buried tunnel in different geological environments is presented and the results are assessed. A schematic model of a tunnel in Tehran-Pardis highway is also constructed and the results of the simulation of the tunnel are compared with the field data.

The Bakhtiari dam has located on Bakhtiari River in province of Lorestan. In order to access the crest of the dam, the excavation of a spiral tunnel is being studied. There are other access tunnels which are branched from this tunnel in different levels and are connected to grout galleries. According to the fact that this tunnel will also be used during the operation of the dam,The correct determination of mechanical parameters of rock masses for tunnel design and stability Analysis is very important. In order to analyse the stability of the underground rock structures, the mechanical and engineering parameters of the rock mass must be known. Accurate rock mass properties can only be obtained from large in situ tests. Such tests are seldom carried out as they are very expensive and time consuming. Sensitivity analysis of parameters can be applied for the optimisation of testing schemes. Sensitivity analysis helps to avoid mistakes due to subjective conjecture. In this article, after the introduction of regional geology and determination critical section on the tunnel path, the mechanical parameters of the rock mass surrounding the tunnel are modelled and analyzed by using FLAC3D software (numerical finite difference method). Parameters conducted in the analysis include the elasticity modulus (E), cohesion of the rock mass (C), friction angle (ϕ), coefficient of lateral stress (K) and tensile strength (&sigmat). Ultimately, according to the result of numerical modelling and parametric analysis, parameters affecting the stability are prioritized. The result of analysis showed that in this project, tensile strength of the rock mass does not affect the stability of the tunnel, and Also, in order of priority, E, ϕ, C, k parameters are important in design. The amount of field tests for rock parameters can be rationalised according to their sensitivity factors.

The behaviour of Ground Penetrating Radar (GPR) electromagnetic field can be simulated using Maxwell’s equations and associated boundary conditions. So far a number of numerical methods for modeling GPR data have been proposed including the popular Time Domain Finite Difference (TDFD) technique. The popularity of TDFD is mainly due to being relatively simple to implement، its high flexibility and capability to simulate complex subsurface geology. Also، the TDFD approach is not only conceptually accurate for complex geological models but also enables us to design realistic antenna and to study physical electromagnetic phenomena such as dispersion in electrical properties. Despite having these advantages، the finite difference method has pitfalls such as becoming very time consuming in simulating the most common media especially with high dielectric permittivity causing the forward modeling process to become very time consuming even by modern high-speed computers. Synthetic GPR responses are useful for predicting expected GPR data over known geometries such as horizontal cylinders and prisms. The GPR data and subsurface lithological and hydrogeological properties are related through a numerical forward modeling engine. Therefore، the GPR forward modeling engine can transform the subsurface electrical properties into expected GPR responses which in turn can be used for optimizing data acquisition procedures on pre-defined subsurface targets. Since any efficient inversion routine requires a fast forward modeling engine، this study aimed the development of a fast forward modeling algorithm capable of being implemented in any inversion routine. To have efficient numerical forward modeling algorithm، we have adapted a leap-frog، staggered-grid approach introduced by Yee، which incorporates offsetting the electric and magnetic field components in both space and time in such a way that the finite difference approximations of the governing partial derivatives in each equation are centered on the same spatiotemporal location. Obviously 2-D modeling is limited and cannot fully account for antenna behaviour and out-of-plane variations in material properties; however، many important features common to most GPR responses can be identified via employing a computationally cost effective 2D algorithm. In the current study، the patterns of GPR responses that are well known to be hyperbola in shape are used as leading models in order to reduce the execution time. A GPR system collects the reflected pulses coming from different depths in the form of traces which when gathered along with a profile، they make a GPR section called radargram. In general، the simulated GPR traces of common reflected objects are time shifted like the Normal MoveOut (NMO) traces encountered in seismic reflection responses. This property suggests the application of Fourier transform to the GPR traces and the use of time shifting property of such transformation to interpolate traces between the adjusted traces in frequency domain. Therefore، the lateral resolution of GPR traces computed along with any profile is enhanced using a linear interpolation in the Fourier domain resulting in an increased speed of the forward modeling algorithm. Selecting the minimum lateral trace to trace interval with the appropriate sampling frequency of the signal، prevent any aliasing to occur. It is shown that such methodology can significantly decrease the computing time by more than 12. 5 times.

The basic point in geophysical studies is to obtain a model that can show structures belowthe ground surface. The basic step for this project is to obtain measurable parameters at the surface of the earth by physical laws. This object can be done by forward modeling. In geophysical investigations، the simplest model representing the earth is a one dimensional model which in the conductivity varies just with the variation of depth. In more realistic models of the earth، resistivity varies not only with depth، but also with one horizontal direction. Two dimensional forward modeling can be solved by analytical methods for some special cases. In two or three dimensional general cases، modeling performs by numerical methods. The numerical method in this case is the finite difference. When the results are extensive، this method is more useful، in comparison with the other calculative computer methods. Finite difference method does need much volume of CPU; also this method is suitable for running on some computers simultaneously because only the nearest adjacent points involve calculations. The main point in this method is the space size of the model blocks and the number of required frequencies. This will determine the duration of running time of the code. Solving the problem of electromagnetic induction in the earth by numerical methods has been considered by many earth scientists for years. Various methods and techniques have been used in this field such as transmission line analogy، integral equation، finite element and finite difference. (Brewitt-Taylor and weaver،1976.) Jones and Pascoe are one of the first persons who have worked in this field. Jones and Pascoe’s articles in 1971 and 1972 further more than theoretical solution of electromagnetic induction، also included FORTRAN codes of electromagnetic induction solution by finite difference method for two dimensional structures in a layered earth. Nowadays researches in this field are focused on solution of electromagnetic induction in three dimensions. An economical and good formulation of this point in Integral vector equation method is presented by Raiche in 1974 and also the same method in more details is presented by Weidelt in 1975 separately. Maxwell equations are needed for forward modeling of magnetotelluric data. The main equations which have been used in this research are curly differential Maxwell equations of electric and magnetic fields and the finite difference method is used for these equations. After setting the preconditions for having softer variations in electrical resistivity، a model for electrical resistivity has been gained in node points. By determination of final formulas of finite difference for all of the blocks of the net، the code for forward modeling in MATLAB is written. MATLAB code after obtaining the values of electric and magnetic fields for all points of the net، determines the electrical resistivity and phase of the points. Eight matrices are the initial data for MATLAB code. These matrices introduce the model parameters. After making the model، MATLAB code makes the matrices of the system equations coefficients for obtaining the electric and magnetic fields values of the net. In conclusion، by calculating the electromagnetic field values، the values of the electrical resistivity and phase are gained for all points. In this study، a forward modeling of TM mode magnetotelluric data is used. The MATLAB code is able to forward model the magnetotelluric data for various artificial two-dimensional models. Three models and their responses are presented. The code performs the forward modeling of various models perfectly.

Usually, simplified models, such as shallow water model, are used to describe atmospheric and oceanic motions. The shallow water equations are widely applied in various oceanic and atmospheric extents. This model is applied to a fluid layer of constant density in which the horizontal scale of the flow is much greater than the layer depth. However, the dynamics of a two-dimensional shallow water model is less general than three-dimensional general circulation models but is preferred because of its greater mathematical and computational simplicity. Taking intrinsic complexity of fluids, recently, numerical researches have been focused on highly accurate methods. Especially, for large grid spacing numerical simulation, the use of highly accurate methods have become urgent. This trend led to an interest in compact finite difference methods. The compact finite-difference schemes are simple and powerful ways to reach the objectives of high accuracy and low computational cost. Compared with the traditional explicit finite-difference schemes of the same-order, compact schemes have proved to be significantly more accurate along with the benefits of using smaller stencil sizes, which can be essential in treating nonperiodic boundary conditions. Application of some families of the compact schemes to the spatial differencing in some idealized models of the atmosphere and oceans shows that compact finite difference schemes can be considered as a promising method for the numerical simulation of geophysical fluid dynamics problems. In this research work, the sixth-order combined compact (CCD6) finite difference method was applied to the spatial differencing of f-plane shallow-water equations in vorticity, divergence and height forms (on a Randall's Z grid). The second-order centered (E2S), fourth-order compact (C4S) and sixth-order super compact (SCD6) finite difference methods were also used for spatial differencing of the shallow water equations and the results were compared to the ones from a pseudo-spectral (PS) method. A perturbed unstable zonal jet was considered as the initial condition for numerical simulation in which it breaks up into smaller vortices and becomes very complex. The shallow water equations are integrated in time using a three-level semi-implicit formulation. To control the build-up of small-scale activities and thus potential for numerical nonlinear instability, the non-dissipative vorticity equation was made dissipative by adding a hyperdiffusion term. The global distribution of mass between isolevels of the potential vorticity, called mass error, was used to assess numerical accuracy. The CCD6 generated the least mass error among finite difference methods used in this research. By taking the PS method as a reference, the qualitative and quantitative comparison of the results of the CCD6, SCD6, C4S and E2S, indicated the high accuracy of the sixth-order combined compact finite difference method.

In many numerical simulations of fluid dynamics problems, especially those possessing a wide range of length and time scales (e.g., geophysical flows), low-order numerical schemes are insufficient. Compact finite difference schemes, introduced as far back as the 1930s, have been found to be simple ways of reaching the objectives of high accuracy and low computational cost. Compared with the traditional explicit finite difference schemes of the same order, the compact schemes are more accurate with the added benefit of using smaller stencil sizes, which can be essential when treating non-periodic boundary conditions. In recent years, the number of studies devoted to the application of compact schemes to spatial differencing of geophysical fluid dynamics problems has been increasing. This work focuses on the application of a three-point fourth-order compact finite difference scheme for numerical solutions of bottom gravity current over a slope. The governing equations used to perform the numerical simulation are the vorticity-stream function-salinity formulation of the two dimensional viscous incompressible Boussinesq equations. The details of spatial and temporal discretization of the governing equations are presented. For spatial differencing of the equations, the second-order central and a three-point fourth-order compact finite difference schemes are employed. In addition, the second-order two-stage predictor-corrector leapfrog scheme is used to advance the governing equations in time. Derivation of the consistent boundary condition formulation to generate stable numerical solution without degrading the global accuracy of the computations is also presented. To derive the required numerical boundary conditions for salinity and vorticity fields at lateral, top and bottom boundaries of the computational domain, the fourth-order one-sided (forward and backward) compact relations are used. Two values for the salinity and a fixed value for bottom slope angle are used to perform the numerical simulations. Qualitative comparison of the results indicates better performance of the fourth-order compact scheme with respect to the second-order method. Furthermore, the computed value of the rate of the head growth of the gravity current generated by the fourth-order compact scheme is in agreement with existing numerical results, which indicates the accuracy of simulations in a quantitative manner. For the test cases used to perform the simulations in the present work, it was observed that the values of salinity and vorticity generated by the second-order method on bottom boundary were too noisy. While, values of salinity and vorticity generated by the fourth-order compact scheme, especially on the bottom boundary of computational domain, do not show this property and are more accurate than those generated by the second-order method. In addition, the numerical results show that the fourth-order compact scheme can successfully simulate the formation of vortices in the tail section of the gravity current, while the second-order scheme fails.

Seismic wave propagation modeling is helpful in understanding waveform behavior involving velocity anomalies due to complicated geological structures. On the other hand, forward modeling is the kernel of inversion and tomography procedures, so developing forward modeling algorithms is an inevitable need. In order to model wave propagation, it is necessary to solve complete wave equation for 3D media. Because of limitations in computational resources, solving complete wave equation with all ideal considerations such as heterogeneity, anisotropy, and absorption, for 3D media is difficult. Thus by considering easier conditions we can solve wave equation with normal computers. Solving wave equation with computers needs to discretization techniques such as boundary integral, finite-element or finite-difference methods. Boundary integral methods are suitable for simple models while finite-element or finite-difference methods are predominantly used for heterogeneous models. Depending on the domain for which wave equation is going to be solved, we can categorize methods to time-space, frequency-space, Laplace, slowness-space and etc. Recently, the frequency domain finite-difference (FDFD) method has found extensive application in multi-source experiment modeling, especially in waveform tomography. Because of the special form of wave equation in the frequency domain, the modeling of multi-source experiments is a straightforward job. On the other hand, considering absorption mechanisms is easy. This study deals with wave propagation modeling in a 2D acoustic heterogeneous media, using the second-order approximation of acoustic wave equation in the frequency domain. The acoustic wave equation is formulated as the first-order hyperbolic system involving fields of pressure and particle velocities. This system is discretized using the second-order staggered grid stencil. To avoid spurious reflections from the model boundaries, a sponge-like perfectly matched layer (PML) is implemented. Solving wave equation in the frequency domain leads to a large matrix equation. The key step in the frequency domain finite-difference modeling that controls computational efficiency is the numerical inversion of the massive matrix. The matrix structure depends on the spatial derivatives approximation. In order to solve this system, different direct solvers can be used. The UMFPACK (unsymmetric multifrontal sparse LU factorization package) solver, which is embedded in MATLAB and has acceptable performance in solving the general system of equations, was selected for this study. For each frequency component, it is necessary to solve a large system of equations to obtain a single frequency component of the pressure wavefield. In this paper we review criteria to avoid numerical dispersion and errors during the finite-difference approximation, and time aliasing during frequency sampling. Choosing appropriate spatial and frequency discretization intervals is very important in FDFD modeling. Choosing large Δ (spatial discretization interval) values will cause the pressure field to be inadequately sampled in space and numerical dispersion. For the scheme presented here, we should have more than 10 grid points per minimum wavelength to keep dispersion errors small. On the other hand according to the sampling theorem, if, where tmax is maximum time and df is frequency components sampling interval, we encounter time aliasing. The present FDFD package is written in MATLAB programming language. MATLAB supports sparse algebra and makes possible the solution of large system of equations with minimum usage of memory, which is the main concern in FDFD algorithms. Examples of waveform modeling using FDFD are shown. In the first example, the headwave originating from a high velocity layer is modeled; in the second example, the wave behavior in a trap model is shown; and in the last example, a salt dome model is studied.