High-resolution satellite images are extensively used in different fields. Geo-referencing process, as innate part of extraction of topographic terrains through these images, has been studied in many researches. In geo-referencing of satellite images, different models can be used, but rational functions are the most suitable options. Determining co-efficiency of rational functions is ill-condition problem, so to solve this problem Tikhonov Regularization method has been used. In such Regularization method, Regularization selection Parameter is very important. In present study, this Parameter was calculated through two methods including: minimizing root mean square of errors (RMSE) and the L-curve for determining co-efficiency of rational functions. Then these two methods have been used in least standard squares of parametric model. Also combined model has been used to determine co-efficiency of rational functions in geo-referencing process. These calculations have been done for two different control-points groups with various numbers and accuracies. Using these two models (parametric and combined), Regularization Parameter has been calculated through L-curve and root mean square of error methods by 55 points. The results show that the root mean square errors and L-curve methods in parametric model led to accuracy of 4.45 and 5.40 pixels, respectively. Also in the combined model, root mean square errors and L-curve methods showed accuracy of 3.42 and 5.10 pixels, respectively. Above calculations were repeated with 120 points. This time, results show approximately same accuracies for both root mean square errors and L-curve methods.

Inversion method is very common in the interpretation of practical gravity data. The goal of 3D inversion is to estimate density distribution of an unknown subsurface model from a set of known gravity observations measured on the surface. The Regularization Parameter is one of the effective Parameters for obtaining optimal model in inversion of the gravity data for similar inversion of other geophysical data. For estimation of the optimum Regularization Parameter the statistical criterion of Akaike’ s Bayesian Information Criterion (ABIC) usually used. This Parameter is experimentally estimated in most inversion methods. The choice of the Regularization Parameter, which balances the minimization of the data misfit and model roughness, may be a critical procedure to achieve both resolution and stability. In this paper the Active Constraint Balancing (ACB) as a new method is used for estimating the Regularization Parameter in two-dimensional (2-D) inversion of gravity data. This technique is supported by smoothness-constrained least-squares inversion. We call this procedure “ active constraint balancing” (ACB). Introducing the Lagrangian multiplier as a spatially-dependent variable in the Regularization term, we can balance the Regularizations used in the inversion. Spatially varying Lagrangian multipliers (Regularization Parameters) are obtained by a Parameter resolution matrix and Backus-Gilbert spread function analysis. For estimation of Regularization Parameter by ACB method use must computed the resolution matrix R. The Parameter resolution matrix R can be obtained in the inversion process with pseudo-inverse G􀬾 multiplied by the kernel G. 􀜴 = 􀜩 􀬾 􀜩 (1) The spread function, which accounts for the inherent degree of how much the ith model Parameter is not resolvable, defined as: 􀜵 􀜲 􀯜 = Σ 􀵫 􀝓 􀯜 􀯝 􀵫 1 − 􀝏 􀯜 􀯝 􀵯 􀜴 􀯜 􀯝 􀯇 􀵯 􀬶 􀯝 􀭀 􀬵 (2) where M is the total number of inversion Parameters, 􀝓 􀯜 􀯝 is a weighting factor defined by the spatial distance between the ith and jth model Parameters, and 􀝏 􀯜 􀯝 is a factor which accounts for whether the constraint or Regularization is imposed on the ith Parameter and its neighboring Parameters. In other words, the spread function defined here is the sum of the squared spatially weighted spread of the ith model Parameter with respect to all of the model Parameters excluding ones upon which a smoothness constraint is imposed. In this approach, the Regularization Parameter λ (x, z) is set by a value from log-linear interpolation. log􁈺 λ 􀭧 􁈻 = log􁈺 λ 􀭫 􀭧 􀭬 􁈻 + 􀭪 􀭭 􀭥 􁈺 􀮛 􀱣 􀱗 􀱮 􁈻 􀬿 􀭪 􀭭 􀭥 􁈺 􀮛 􀱣 􀱟 􀱤 􁈻 􀭪 􀭭 􀭥 􁈺 􀭗 􀭔 􀱣 􀱗 􀱮 􁈻 􀬿 􀭪 􀭭 􀭥 􁈺 􀭗 􀭔 􀱣 􀱟 􀱤 􁈻 × 􁈼 log􁈺 SP􀭧 􁈻 − log􁈺 SP􀭫 􀭧 􀭬 􁈻 􁈽 (3) where 􀜵 􀜲 􀯠 􀯜 􀯡 and 􀜵 􀜲 􀯠 􀯔 􀯫 are the minimum and maximum values of spread function 􀜵 􀜲 􀯜 , respectively, and the 􀟣 􀯠 􀯜 􀯡 and 􀟣 􀯠 􀯔 􀯫 are minimum and maximum values of the Regularization Parameter λ (x, z), which must be provided by the user. With this method, we can automatically set a smaller value λ (x, z) of the Regularization Parameter to the highly resolvable model Parameter, which corresponds to a smaller value of the spread function SP􀭧 in the inversion process and vice versa. Users can choose these minimum and maximum Regularization Parameters by setting variables LambdaMin and LambdaMax. For getting the target an algorithm is developed that estimates this Parameter. The validity of the proposed algorithm has been evaluated by gravity data acquired from a synthetic model. Then the algorithm used for inversion of real gravity data from Matanzas Cr deposit. The result obtained from 2D inversion of gravity data from this mine shows that this algorithm can provide good estimates of density anomalous structures within the subsurface.

Gravity data inversion is one of the important steps in the interpretation of practical gravity data. The inversion result can be obtained by minimization of the Tikhonov objective function. The determination of an optimal Regularization Parameter is highly important in the gravity data inversion. In this work, an attempt was made to use the active constrain balancing (ACB) method to select the best Regularization Parameter for a 3D inversion of the gravity data using the Lanczos bidiagonalization (LSQR) algorithm. In order to achieve this goal, an algorithm was developed to estimate this Parameter. The validity of the proposed algorithm was evaluated by the gravity data acquired from a synthetic model. The results of the synthetic data confirmed the correct performance of the proposed algorithm. The results of the 3D gravity data inversion from this chromite deposit from Cuba showed that the LSQR algorithm could provide an adequate estimate of the density and geometry of sub-surface structures of mineral deposits. A comparison of the inversion results with the geologic information clearly indicated that the proposed algorithm could be used for the 3D gravity data inversion to estimate precisely the density and geometry of ore bodies. All the programs used in this work were provided in the MATLAB software environment.

In this paper a fast method for large-scale sparse inversion of magnetic data is considered. The L1-norm stabilizer is used to generate models with sharp and distinct interfaces. To deal with the nonlinearity introduced by the L1-norm, a model-space iteratively reweighted least squares algorithm is used. The original model matrix is factorized using the Golub-Kahan bidiagonalization that projects the problem onto a Krylov subspace with a significantly reduced dimension. The model matrix of the projected system inherits the ill-conditioning of the original matrix, but the spectrum of the projected system accurately captures only a portion of the full spectrum. Equipped with the singular value decomposition of the projected system matrix, the solution of the projected problem is expressed using a filtered singular value expansion. This expansion depends on a Regularization Parameter which is determined using the method of Generalized Cross Validation (GCV), but here it is used for the truncated spectrum. This new technique, Truncated GCV (TGCV), is more effective compared with the standard GCV method. Numerical results using a synthetic example and real data demonstrate the efficiency of the presented algorithm.

In this paper the inversion of gravity data using L1–norm stabilizer is considered. The inversion is an important step in the interpretation of data. In gravity data inversion, the goal is to estimate density and geometry of the unknown subsurface model from a set of known observation measured on the surface. Commonly, rectangular prisms are used to model the subsurface under the survey area. The unknown density contrasts within each prism are the Parameters which should be estimated. The inversion of gravity data is an example of underdetermined and ill-posed problem, i.e. the solution can be non-unique and unstable. Thus, in order to find an acceptable solution Regularization should be imposed. Solution is usually obtained by minimizing a global objective function consisting of two terms, data misfit and the Regularization term. Data misfit measures how well an obtained model can reproduce the observed data. Usually, it is assumed noise in gravity data is Gaussian, therefore a L2–norm measure of the error between observed and predicted data is well suited for data misfit. There are several choices for a stabilizer, depends on type of features one wants to see from inverted model. A typical choice is a L2 –norm of a low-order differential operator applied to the model, which also a priori information and depth weighting can be incorporated (Li and Oldenburg, 1996). In this case the objective function is quadratic, then minimization of the function results a linear system to be solved. However, the models recovered in this way are characterized by smooth feature which are not always consistent with the real geological structures. There are situations in which the sources are localized and separated by sharp, distinct interfaces. To deal with this problem, during last decades, researchers have proposed a few types of stabilizer. Last and Kubik (1983) presented a compactness criterion for gravity inversion that seeks to minimize the area (or volume in 3D) of the causative body. Portniaguine and Zhdanov (1999) based on this stabilizer, who named the minimum support (MS), developed the minimum gradient support (MGS) stabilizer. For both constraint, the Regularization term can be written as the weighted L2–type norm of the model. Therefore, the problem of the minimization of the objective function can be treated same as conventional Tikhonov functional. The only difference is that a priori variable weighting matrix for model Parameters incorporated in the Regularization term. Thus the Iteratively Reweighted Least Square (IRLS) algorithm is required to solve the problem. Other possibility for stabilizer is the minimization of the L1-norm of model or gradient of model, the latter indicates total variation Regularization. The L1–norm stabilizer allows occurrence of large elements in the inverted model among mostly small values. Therefore, it can be used to obtain sharp boundaries and blocky features. Although the L1–norm stabilizer has favorable properties, in reconstruction of sparse models, its numerical implementation in a minimization problem can be difficult because its derivatives with respect to an element is not defined at zero. To overcome this difficulty, in this paper, the L1–norm stabilizer is approximated by a reweighted L2 –norm term. The algorithm is extended to gravity inverse problem, which needs depth weighting and other priori information to be included in the objective function. For estimating the Regularization Parameter, which balances between two terms of objective function, the Unbiased Predictive Risk Estimator (UPRE) method is used. The solution of the resulting objective functional is found using Generalized Singular Value Decomposition (GSVD), also provides for efficient determination of the Regularization Parameter at each iteration. Simulation using synthetic data of a dipping dike demonstrates that the method is capable to reconstruct focused image, boundaries and slop of the reconstructed model are close to those of the original model. The method is applied on gravity data acquired over the Gotvand dam site, in the south-west of Iran. The results show rather good agreement with those obtained from the boreholes.

Inversion of magnetic data is one of the important steps in the interpretation of practical magnetic data. The inversion result can be obtained by minimization of Tikhonov objective function. The determination of an optimal Regularization Parameter is highly important in magnetic data inversion. In this paper, an attempt has been made to use unbiased predictive risk estimator (UPRE) method in selecting the best Regularization Parameter for 3D constrained inversion of magnetic data using gradient projection reduced Newton (GPRN) algorithm. To achieve this goal, an algorithm has been developed to estimate this Parameter. The validity of the proposed algorithm has been evaluated by magnetic data acquired from a synthetic model. The results have been compared with the results of generalized cross validation (GCV) method. The GCV method failed to estimate the Regularization Parameter, but the UPRE method could find the best Regularization Parameter. Then, the algorithm was used for inversion of real magnetic data obtained from Allah Abad iron deposit. The results of three-dimensional (3-D) inversion of magnetic data from this iron deposit show that the GPRN algorithm can provide an adequate estimate of magnetic susceptibility and geometry of subsurface structures of mineral deposits. A comparison of the inversion results with drilling data clearly indicate that the proposed algorithm can be used for 3-D inversion of magnetic data to estimate precisely the magnetic susceptibility and geometry of magnetized ore bodies.

One of the most important social problem after world warII in developing country is rapid urbanization. In most of the developing country yearly growth urban population is among 5 untill 8 percent. This urban explosive grow and the slums are effect of the inside immigration from village to urban, that call in various place and various form like marginal, squatter, Illegal, Irregular, spontaneous, unauthorized, informal settlement.Our country (IRAN) like other developing country encounter with this problem. At present in most urban and industrial city like Arak city there are slums.These thesis investigate structural and cultural feature of slums (Bagh Khalaj district) that doing with document and surveying at first abstract of science literature (definitions, scores and feature) of slums in other country and Iran and then survey physical and humanly structure of Arak city.After that survey slums and especially Bagh Khalaj district (case study) in Arak city. At the end of thesis on the base of results and to point out strategy, limitations, facilities and problems and then present solutions.

In this paper the 3D inversion of gravity data using two different Regularization methods, namely Tikhonov Regularization and truncated singular value decomposition (TSVD), is considered. The earth under the survey area is modeled using a large number of rectangular prisms, in which the size of the prisms are kept fixed during the inversion and the values of densities of the prisms are the model Parameters to be determined. A depth weighting matrix is used to counteract the natural decay of the kernel, so the inversion obtains reliable information about the source distribution with respect to depth. To generate a sharp and focused model, the minimum support (MS) constraint is used, which minimizes the total area with non zero departure of the model Parameters from a given a priori model. Then, the application of iteratively reweighted least square algorithm is required to deal with non-linearity introduced by MS constraint. At each iteration of the inversion, a priori variable weighting matrix is updated using model Parameters obtained at the previous iteration. We use the singular value decomposition (SVD) for computing Tikhonov solution, which also helps us to compare the results with the solution obtained by TSVD. Thus, the algorithms presented here are suitable for small to moderate size problems, where it is feasible to compute the SVD. In Tikhonov Regularization method, the optimal Regularization Parameter at each iteration is obtained by application of the x2 - principle Parameter-choice method. The method is based on the statistical distribution of the minimum of the Tikhonov function. For weighting of the data fidelity by a known Gaussian noise distribution on the measured data and, when the Regularization term is considered to be weighted by unknown inverse covariance information on the model Parameters, the minimum of the Tikhonov functional becomes a random variable that follows a x2 distribution. Then, a Newton rootfinding algorithm can be used to find the Regularization Parameter. For truncated SVD Regularization, the Picard plot is used to find a suitable value of truncation index. In math literature, a plot of singular values together with SVD and solution coefficients is often referred to as Picard plot. To test the algorithms, a density model which consists of a dipping dike embedded in a uniform half-space is used. The surface gravity anomaly produced by this model is contaminated with three different noise levels, and are used as input for introduced inversion algorithms. The results indicate that the algorithms are able to recover the geometry and density distribution of the original model. In general, the reconstructed model is more sparse using TSVD method as compare with Tikhonov solution. This especially happens for high noise level, where there is an important difference between two solutions. In this case, while TSVD produces a sparse model, the solution of Tikhonov Regularization is not sparse. Furthermore, the number of iterations, which is required to terminate the algorithms, is more for TSVD as compare with Tikhonov method. This feature, along with automatic determination of Regularization Parameter, makes the implementation of the Tikhonov Regularization method faster than TSVD. The inversion methods are used on real gravity data acquired over the Gotvand dam site in the south-west of Iran. Tertiary deposits of the Gachsaran formation are the dominant geological structure in this area, and it is mainly comprised of marl, gypsum, anhydrite and halite.There are several solution cavities in the area so that relative negative anomalies are distinguishable in the residual map. A window of residual map consists of 640 gridded data, which includes three negative anomalies, that is selected for modeling. The reconstructed models are shown and compare with results obtained by bore holes.

Ill-conditioned linear systems may frequently arise in discretization of integral equations and many other real world applications. Solving such systems by classical methods might fail or result to solutions that are meaningless from practical point view. Moreover a slight perturbation in the right hand side vector might also lead to an enormous change of the solution vector due to ill-conditioned ness. To find meaningful solutions of such systems, the Tikhonov Regularization is an effective technique that has been widely used. In this paper we use it to solve ill-conditioned linear systems and also to find closest feasible linear systems to nearly feasible linear systems by smallest changes in problem data. Throughout the paper numerical results are reported to demonstrate the practical efficiency of the presented algorithms.