HEAT TRANSFER MODELING IN EARTH SCIENCES: STEADY-STATE CONDUCTIVE HEAT TRANSFER

Sayısal modelleme bilim ve mühendisliğin pek çok alanında yaygın bir şekilde kullanılmaya başlanmıştır. Bu çalışmada da iki boyutlu kararlı-hal kondüktif ısı transferi problemlerinin yerbilimleri disiplini içindeki uygulamalarının temelleri üzerinde durulmuş ve örnekler verilmiştir. Problemler sonlu farklar yaklaşımıyla relaksasyon (succesive overrelaxation) yöntemi kullanılarak çözülmüştür. Modellemelerde ısı üreten kaynakların yer içinde üstel olarak azaldığı kabul edilmiştir. Örnekler varsayımsal bir kabuk modelindeki ısı transferi ile graben türü bir yapıdaki ısı transferi problemlerini içermektedir. Sayısal modelleme yerbilimlerindeki problemlerin çözümünde ve yer içindeki fiziksel olayların nasıl geliştiğinin anlaşılmasında önemli katkılar sağlayacak bir araç olarak gözükmektedir. Numerical modeling has been widely used in the fields of science and engineering. This study outlines the use of two dimensional steady-state conductive heat transfer modeling in earth sciences and gives some examples. The problems were solved by the successive over relaxation method of finite differences. Exponential decrease with depth in heat sources in the crust was assumed as heat production model. Examples include heat transfer problems in a hypotetical crust and graben models. It seems that numerical modeling is a useful tool for solving problems in earth sciences and understanding mechanisms of physical processes inside the earth

A hybrid approach for tomographic inversion of crosshole seismic first-arrival times

A sequential hybrid approach was presented here to invert crosshole seismic first-arrival times. The proposed tomographic scheme combined a simple simulated annealing algorithm with a linearized smoothness-constrained least-squares inversion. The simulated annealing was implemented to obtain a background velocity distribution used by the linearized inversion for the initial guess. The linearized component was based on the functional description of traveltimes. This indicates a nonlinear function, the eikonal equation, providing traveltimes for a given slowness model. Thus an explicit ray tracing was not required by the linearized scheme. The velocity updates were obtained by a matrix inversion based on an iterative conjugate gradient-like LSQR algorithm. Second-difference regularization was used to stabilize the solutions. The Jacobian matrix giving the partial derivatives with respect to the model parameters was constructed by a finite-difference approximation based on the perturbation of the cell slowness. The traveltimes for both the hybrid and linearized schemes were calculated by a fast finite-difference eikonal solver. The hybrid scheme was tested by using both synthetic and field data sets based on the crosshole geometry. According to the tests studies, the tomograms resulted from the hybrid approach better imaged the subsurface velocity distribution. Also the hybrid optimization was characterized by quicker convergence rate than the conventional optimization based on only the linearized inversion. The tests with the synthetic data set also showed that the hybrid approach yielded a solution having lower rms residual, smaller Euclidean distance and lower relative errors in the cell velocities

Seismic first-arrival tomography with functional description of traveltimes

A two-dimensional smoothness-constrained least-squares inversion scheme was applied to seismic first-arrival time data. The inversion scheme was based on the functional description of traveltimes; thus, it did not require a step of ray tracing, and traveltimes were obtained by a finite-difference eikonal solver. The Laplacian difference of cell slownesses was used for the smoothness constraint. Model velocities were obtained by matrix inversion including the QR decomposition and iterative LSQR method. The Jacobian matrix of the partial derivatives was constructed by a finite-difference approximation based on the perturbation of the cell slowness. Since the construction of the Jacobian matrix was the most time-consuming step of the inversion scheme, Broyden's update was used for this matrix, and it was replaced by its numerical approximation obtained from Broyden's method after the third iteration and for all subsequent iterations to expedite the inversion process. This significantly improved the computational performance of the scheme by reducing the computer time for the calculation of the Jacobian matrix. The algorithm was tested by using a number of synthetic and field data sets. The test studies included both surface seismic refraction and crosshole seismic data acquisition configurations. Also image appraisal analyses were performed for the solutions obtained from both surface and crosshole field data sets by calculating model covariance and model resolution matrices. The presented algorithm yielded satisfactory results during the test studies. The stability and fast convergence rate were the main characteristics of the algorithm

Upper crustal P-wave velocity structure of Kii Peninsula, SW Japan by first-arrival traveltime inversion of the Kawachinagano-Kiwa refraction profile

We investigated the upper crustal P-wave velocity structure down to 5 km in the Kii Peninsula, SW Japan by a tomographic inversion of the first-arrival traveltimes from the seismic refraction experiment performed by the Research Group for Explosion Seismology (RGES) in 1988. The observations were carried out on a profile running in the N-S direction from Kawachinagano, Osaka Prefecture to Kiwa, Mie Prefecture. The profile extends for about 65 km across the major geological zones, which characterize the geological features of SW Japan. Six shots were fired and the generated seismic waves were recorded at 86 temporary observation sites. The inversion scheme was applied to 359 first-arrival times to delineate the velocity structure along the profile. In the tomographic scheme, velocity estimation was achieved by an iterative, linearized least-squares inversion. The Jacobian matrix was constructed via a finite-difference approximation by perturbing the slownesses of the cells instead of performing ray tracing. Traveltime calculations were carried out by using a fast finite-difference eikonal solver. Velocity updates were obtained by a matrix inversion algorithm using a conjugate gradient least-squares scheme. In addition, model covariance and model resolution matrices were obtained to assess the velocity image. Two low-velocity zones observed in the northern and southern parts of the profile are the most prominent features of the tomogram. The P-wave velocity structure is generally consistent with the surface geology, and the fault zones associated with the Median (MTL) and Gobo-Hagi (GHTL) tectonic lines across the peninsula

Traveltime tomography of crosshole radar data without ray tracing

We presented a least-squares traveltime inversion algorithm for crosshole ground-penetrating radar (GPR) direct-arrival data. The proposed scheme used the eikonal equation as traveltime functional and thus avoided tracing rays during inversion. The Jacobian matrix is constructed by a finite-difference approximation via the perturbation of slowness. The solutions were obtained by an iteratively linearized inversion approach. A smoothness-type regularization was implemented to stabilize the solutions. Traveltime calculations in forward modeling were performed by a finite-difference eikonal solver that allows modeling wavefronts. Matrix inversions were achieved by using conjugate gradient least-squares (CGLS) and LSQR algorithms. Broyden's method was used to accelerate the calculation of the Jacobian matrix when the number of model parameters was large. We tested the proposed method on three synthetic data sets and on a field data set from the Boise Hydrogeophysical Research Site (BHRS), Idaho; and we compared our model for the field data with the one obtained by a ray-tracing-based algorithm. This comparison indicated that the suggested inversion scheme was able to generate a solution as good as the one resulting from a conventional ray-based scheme. The synthetic data were obtained from simple to complex subsurface velocity distributions, including low- and high-velocity anomalies. Additionally, an image quality analysis was performed by calculating model covariance and model resolution matrices for one of the synthetic models having a complex subsurface structure and for the model resulting from the field data. All inversions were characterized by fast and stable convergence. Tests with noisy data sets indicated that the tomograms were relatively insensitive to noise in the data. It was also observed that the LSQR algorithm produced better results than the CGLS did in the tests with the synthetic models having complex subsurface structures. We considered the proposed technique to be an efficient traveltime inversion scheme for crosshole radar data. (C) 2010 Elsevier B.V. All rights reserved

Inversion of self-potential anomalies caused by simple-geometry bodies using global optimization algorithms

Three naturally inspired meta-heuristic algorithms-the genetic algorithm (GA), simulated annealing (SA) and particle swarm optimization (PSO)-were used to invert some of the self-potential (SP) anomalies originated by some polarized bodies with simple geometries. Both synthetic and field data sets were considered. The tests with the synthetic data comprised of the solutions with both noise-free and noisy data; in the tests with the field data some SP anomalies observed over a copper belt (India), graphite deposits (Germany) and metallic sulfide (Turkey) were inverted. The model parameters included the electric dipole moment, polarization angle, depth, shape factor and origin of the anomaly. The estimated parameters were compared with those from previous studies using various optimization algorithms, mainly least-squares approaches, on the same data sets. During the test studies the solutions by GA, PSO and SA were characterized as being consistent with each other; a good starting model was not a requirement to reach the global minimum. It can be concluded that the global optimization algorithms considered in this study were able to yield compatible solutions with those from widely used local optimization algorithms

Modeling of crosshole ground-penetrating radar data

The ground-penetrating radar (GPR) that is one of the non-invasive electromagnetic methods of applied geophysics is widely used to image shallow subsurface with extremely high resolution. The resolution and depth being two important aspects in a GPR survey are affected by the water, clay, soluble salt contents of soils and the center frequency of antenna. It may be difficult to obtain a good subsurface image at desired resolution and targeted depth in the areas characterized by high electrical conductivity. Therefore, a GPR survey based on the crosshole configuration can be a good alternative approach to achieve more detailed subsurface radar velocity distribution. In this study, firstarrival traveltimes being essential for tomographic inversion of crosshole GPR data sets were calculated by a finite-difference timedomain (FDTD) solutions of Maxwell's equations and finite-difference solution of the Eikonal equation throughout a gridded velocity field. Two theoretical subsurface models were used in modeling. In the first model, the subsurface divided into two layers. The second model includes low-and high-velocity blocks embedded in a homogenous medium. The effect of ground-air interface in modeling and the importance of the ratio between separation and depth of boreholes in a crosshole radar survey were also shown during the test studies. Radargrams consisting of the vertical component of the electric field (Ez) recorded in time at the entire receiver locations were acquired from FDTD modeling. Traveltime contour maps for source locations with different depths were obtained from a fast finite-difference Eikonal solver. Raypaths having the minimum traveltime were then calculated by following the steepest gradient direction from the receiver to the transmitter. As a result, the first-arrival traveltimes obtained from both modeling approaches are quite compatible with each other. FDTD modeling is an important tool to determine and evaluate of the wave phases corresponding to the first arriving wave. On the other hand, Eikonal-equation-based modeling presents an approach being highly effective for directly computing first-arrival traveltimes