A Constrained Nonlinear Inversion Approach to Quantitative Interpretation of Self-potential Anomalies Caused by Cylinders,Spheres and Sheet-like Structures

An interpretative method based on a nonlinearly mathematical optimization concept has been developed in this paper, in order to interpret self-potential anomalies (SP) due to horizontal cylinder, vertical cylinder, sphere and sheet-like structures. This interpretative method comprises three main steps. The first step is to formulate mathematically a nonlinearly constrained minimization problem (NCMP) to describe the geophysical problem related to the studied structure. The second one is to suggest an interior penalty function in order to convert the nonlinearly constrained minimization problem (NCMP) into a nonlinearly unconstrained minimization one (NUMP). The third step is to solve the converted nonlinearly unconstrained minimization problem (NUMP) by the well-known Hooke and Jeeves direct search algorithm in order to estimate the geophysical parameters of the studied structure, i.e., depth, polarization angle, electric dipole moment (magnitude of polarization) and geometric shape factor. The Hooke and Jeeves direct search algorithm is purposely chosen for being robust and its application to SP data allows a rapid convergence towards the optimal estimate of parameters. This interpretative method was first tested on theoretical synthetic models with different random noise, where a very close agreement was obtained between assumed and evaluated parameters.The validity of the proposed interpretative method is also tested on practical field examples taken from Turkey, India and Germany, where available SP data existed and was previously analyzed by different interpretative methods. The agreement between the results obtained by the developed method and those obtained by other published methods is good.Acknowledgment Authors would like to thank Dr. I. Othman Director General of the Atomic Energy Commission of Syria for his interest and continuous encouragement to achieve this work. Special thanks to the reviewers for their constructive suggestions aimed at enhancing the quality of this paper.     

On the boundary integral equations in the theory of elastic vibrations

Forward seismic problems are solved for elastic media by rigorous methods (i.e., methods with controllable accuracy). Analysis of the current state of research on this subject suggests that the most promising methods are based on integral and integro-differential equations, notwithstanding the rather modest results of their application to solving forward problems in the theory of elastic vibrations. The second Green integral theorem for seismic waves, formulated and proven in the paper, yields a system of two boundary (surface) integral equations for the displacement vector u(M ₀) and the normal (to the boundary surface) vector component of the stress tensor t_n(M ₀). The integrands of the surface integrals in terms of which the function t_n(M ₀) is expressed on both sides of the interface between the medium and the heterogeneity contain the second derivatives of the Green’s tensor functions ? ^e (M ₀, M) and ? ⁱ (M ₀, M), respectively, which are responsible for a cubic singularity (third-order singularity) if the integration point M coincides with the observation point M ₀. An original method of eliminating the cubic singularity proposed in the paper involves special tensor normalization of the integrals on the outer and inner sides of the interface and subsequent subtraction of one integral from another in order to construct the second integral equation.     

A methodology to select seismic coefficients based on upper bound “Newmark” displacements using earthquake records from Turkey

Strong motion records taken during earthquakes in Turkey are used to calculate Newmark displacements in slopes. These displacements are then utilized in developing a novel displacement-based methodology to select the seismic coefficient which is used to calculate pseudostatic safety factor. In the first step of the study, calculated Newmark displacements are evaluated in three different categories which are as follows: using all data, using data for different earthquake magnitude (M) ranges with and without distance constraint and using data for different peak acceleration (a_max) ranges. For all categories, different equations are obtained to assign slope displacements as a function of the ratio of yield acceleration to peak acceleration. The results show that categorization of data is an important issue, because the displacements are earthquake magnitude and peak acceleration dependent. In the second step, equations obtained for different peak acceleration ranges are used to propose charts linking upper bound slope displacements (D), seismic coefficients (k_h) and pseudostatic safety factors (PSF), which are three important parameters of a pseudostatic approach. This enables the k_h values be chosen based on the allowable displacements, instead of the current applications based on judgement and expertise. The results show that k_h values for any allowable displacement should be based on anticipated a_max values, while use of high PSF values results in lower displacements. Extensive comparison with solutions from the literature is also made. The methodology is best suited for earthquake triggered shallow landslides in natural slopes, consisting of materials which do not lose strength during dynamic loading.     

A model for tides and currents in the English Channel and southern North Sea

The amplitude and phase of 11 tidal constituents for the English Channel and southern North Sea are calculated using a frequency domain, finite element model. The governing equations — the shallow water equations — are modifed such that sea level is calculated using an elliptic equation of the Helmholz type followed by a back-calculation of velocity using the primitive momentum equations. Triangular elements with linear basis functions are used. The modified form of the governing equations provides stable solutions with little numerical noise. In this field-scale test problem, the model was able to produce the details of the structure of 11 tidal constituents including O₁, K₁, M₂, S₂, N₂, K₂, M₄, MS₄, MN₄, M₆, and 2MS₆.     

An efficient numerical model for multicomponent compressible flow in fractured porous media

An efficient and accurate numerical model for multicomponent compressible single-phase flow in fractured media is presented. The discrete-fracture approach is used to model the fractures where the fracture entities are described explicitly in the computational domain. We use the concept of cross flow equilibrium in the fractures. This will allow large matrix elements in the neighborhood of the fractures and considerable speed up of the algorithm. We use an implicit finite volume (FV) scheme to solve the species mass balance equation in the fractures. This step avoids the use of Courant–Freidricks–Levy (CFL) condition and contributes to significant speed up of the code. The hybrid mixed finite element method (MFE) is used to solve for the velocity in both the matrix and the fractures coupled with the discontinuous Galerkin (DG) method to solve the species transport equations in the matrix. Four numerical examples are presented to demonstrate the robustness and efficiency of the proposed model. We show that the combination of the fracture cross-flow equilibrium and the implicit composition calculation in the fractures increase the computational speed 20–130 times in 2D. In 3D, one may expect even a higher computational efficiency.     

半拉格朗日模式风的半解析解及与差分解的比较试验

质点的轨迹计算是半拉格朗日模式的重要基础,传统的数值计算方法由于采用时间差分代替微分,只能得到质点运动轨迹终点的速度,因此质点的移动轨迹（位移）只能靠风速外推的方法计算,导致了模式计算不稳定等问题.借鉴精细积分法中使用半解析解的思路,利用正压原始方程研究了用运动方程的半解析解构建数值模式的可能性.求解了运动方程的一阶和二阶微分方程组的半解析解,通过时间积分半解析解计算质点运动轨迹.数值试验表明,一阶微分方程组的半解析解比差分解略有优势.二阶微分方程组的半解析解在时间步长增大时优势非常明显,而且在保证计算精度的前提下,节省计算时间,这对提高模式性能有重要作用.     

Two-dimensional direct current (DC) resistivity inversion: Data space Occam's approach

A data space Occam's inversion algorithm for 2D DC resistivity data has been developed to seek the smoothest structure subject to an appropriate fit to the data. For traditional model space Gauss–Newton (GN) type inversion, the system of equations has the dimensions of M × M, where M is the number of model parameter, resulting in extensive computing time and memory storage. However, the system of equations can be mathematically transformed to the data space, resulting in a dramatic drop in its dimensions to N × N, where N is the number of data parameter, which is usually less than M. The transformation has helped to significantly reduce both computing time and memory storage. Numerical experiments with synthetic data and field data show that applying the data space technique to 2D DC resistivity data for various configurations is robust and accurate when compared with the results from the model space method and the commercial software RES2DINV.     

A time-splitting pressure-correction projection method for complete two-fluid modeling of a local scour hole

A two-dimensional vertical (2DV), Eulerian two-phase model or complete two-fluid model of the free surface flow was developed to simulate water-sediment flow in a local scour hole. In the model, the complete forms of the vertical, two-dimensional, two-fluid Navier-Stokes equations were discretized using a finite volume scheme. This discretization was done based on a standard staggered grid system using a curvilinear network system in compliance with the bed boundaries and water level. At the beginning of the computational cycle, the equations governing the fluid phase were solved based on the two-step projection method with a pressure-correction technique. In the first step, the intermediate fluid velocities were obtained by solving different phases of the momentum equations of the fluid phase using the time-splitting technique. In the second step, pressure was obtained and fluid velocities were updated. In this step a simple discretization method was applied for decreasing the computational complexity. After obtaining all the fluid phase variables at a new time step, the sediment phase momentum equations were solved using the time-splitting technique and sediment velocities were obtained. Then, at the end of the computational cycle, the sediment phase mass equation was solved and the concentrations of both phases were updated. At last, the capacity of the model for simulating of the longitudinal fluid velocity and sediment concentration in a local scour hole was evaluated. Numerical results were found to be in good agreement with experimental data.     

Separation of Source, Propagation and Site Effects from S Waves¶of Local Earthquakes in Bursa Region, Northwestern Turkey

—?We have used micro-earthquake recordings (M= 1.8–4.1) of local events in the distance range of 5–60?km in order to quantify the attenuation and site effects in the vicinity of the Bursa city, Marmara region, Turkey. The data set consists of 120 three-component recorded accelograms from 69 earthquakes, recorded at six stations. Each station is deployed on different geologic units, such as massive limestone, slope deposit and Quaternary young sediments, in the framework of the Marmara Poly-Project.¶In this study a nonparametric inversion method was applied to acceleration records from the Bursa region to estimate source, site and path effects using a two-step inversion. At the first step, we determined attenuation functions by analyzing the distance dependence of the spectral amplitudes and retrieved values of Q _s (f) = 46.59f ^0.67. At the second step, the corrected S-waves spectral records for the attenuation function, including the geometrical spreading effect, were inverted to separate source and site response for 21 different frequencies selected between 0.5 and ～25?Hz. The near-surface attenuation, κ value, was also estimated by using the model proposed by Anderson and Hough (1984) at each site. We observed that κ₀ is smaller for stations located on rock site (I?dιr, SIGD, κ₀～0.004) compared to the one that is located on Neogene sediment (Çukurca, SCKR, κ₀～0.018).¶Site amplifications from inversion showed that the station located within the Bursa basin, Çukurca (SCKR), is the most important site with about 4.0 amplification value at 1.8?Hz. Demirta? (SDEM) amplifies the spectral amplitudes about 3.0 times at 2.0?Hz, SHMK about 3.5 times between 2.5 and 3.5?Hz and SHMT nearly reaching 3.5 times between 1.5 and 4.0?Hz. However, stations located on the Uluda? Mountain Massif (SKAY and SIGD), which correspond to a deep limestone geological unit, have the smallest amplification, that values between 0.6 and 1.4.     

Rapid source inversions of the 2023 SE Türkiye earthquakes with teleseismic and strong-motion data

We conducted rapid inversions of rupture process for the 2023 earthquake doublet occurred in SE Türkiye, the first with a magnitude of M_W7.8 and the second with a magnitude of M_W7.6, using teleseismic and strong-motion data. The teleseismic rupture models of the both events were obtained approximately 88 and 55 minutes after their occurrences, respectively. The rupture models indicated that the first event was an asymmetric bilateral event with ruptures mainly propagating t...     

Assimilation of altimetry data for nonlinear shallow-water tides: Quarter-diurnal tides of the Northwest European Shelf

Non-linear tidal constituents, such as the overtide M₄ or the compound tide MS₄, are generated by interaction in shallow seas of the much larger astronomically forced “primary” tidal constituents (e.g., M₂, S₂). As such, errors in modeling these “secondary” shallow-water tides might be expected to be caused first of all by errors in modeling the primary constituents. Thus, in the context of data assimilation, observations of primary-constituent harmonic constants can indirectly constrain shallow-water constituents. Here we consider variational data assimilation for primary and secondary tidal constituents as a coupled problem, using a simple linearized perturbation theory for weak interactions of the dominant primary constituents. Variation of the resulting penalty functional leads to weakly non-linear Euler–Lagrange equations, which we show can be solved approximately with a simple two-stage scheme. In the first stage, data for the primary constituents are assimilated into the linear shallow water equations (SWE), and the resulting inverse solutions are used to compute the quadratic interactions in the non-linear SWE that constitute the forcing for the secondary constituents. In the second stage, data for the compound or overtide constituent are assimilated into the linear SWE, using a prior forced by the results of the first stage. We apply this scheme to assimilation of TOPEX/Poseidon and Jason altimetry data on the Northwest European Shelf, comparing results to a large set of shelf and coastal tide gauges. Prior solutions for M₄, MS₄ and MN₄ computed using inverse solutions for M₂, S₂, and N₂ dramatically improve fits to validation tide gauges relative to unconstrained forward solutions. Further assimilation of along-track harmonic constants for these shallow-water constituents reduces RMS differences to below 1 cm on the shelf, approaching the accuracy of the validation tide gauge harmonic constants.     

Solution of the nonlinear transport equation using modified Picard iteration

The transport and fate of reactive chemicals in groundwater is governed by equations which are often difficult to solve due to the nonlinear relationship between the solute concentrations for the liquid and solid phases. The nonlinearity may cause mass balance errors during the numerical simulation in addition to numerical errors for linear transport system. We have generalized the modified Picard iteration algorithm of Celia et al.⁵ for unsaturated flow to solve the nonlinear transport equation. Written in a ‘mixed-form’ formulation, the total solute concentration is expanded in a Taylor series with respect to the solution concentration to linearize the transport equation, which is then solved with a conventional finite element method. Numerical results of this mixed-form algorithm are compared with those obtained with the concentration-based scheme using conventional Picard iteration. In general, the new solver resulted in negligible mass balance errors (< ∥10⁻⁸∥%) and required less computational time than the conventional iteration scheme for the test examples, including transport involving highly nonlinear adsorption under steady-state as well as transient flow conditions. In contrast, mass balance errors resulting from the conventional Picard iteration method were higher than 10% for some highly nonlinear problems. Application of the modified Picard iteration scheme to solve the nonlinear transport equation may greatly reduce the mass balance errors and increase computational efficiency.     

Parallel iterative solution for h and p approximations of the shallow water equations

A p finite element scheme and parallel iterative solver are introduced for a modified form of the shallow water equations. The governing equations are the three-dimensional shallow water equations. After a harmonic decomposition in time and rearrangement, the resulting equations are a complex Helmholz problem for surface elevation, and a complex momentum equation for the horizontal velocity. Both equations are nonlinear and the resulting system is solved using the Picard iteration combined with a preconditioned biconjugate gradient (PBCG) method for the linearized subproblems. A subdomain-based parallel preconditioner is developed which uses incomplete LU factorization with thresholding (ILUT) methods within subdomains, overlapping ILUT factorizations for subdomain boundaries and under-relaxed iteration for the resulting block system. The method builds on techniques successfully applied to linear elements by introducing ordering and condensation techniques to handle uniform p refinement. The combined methods show good performance for a range of p (element order), h (element size), and N (number of processors). Performance and scalability results are presented for a field scale problem where up to 512 processors are used.     

Stability of finite difference numerical simulations of acoustic logging-while-drilling with different perfectly matched layer schemes

In acoustic logging-while-drilling （ALWD） finite difference in time domain （FDTD） simulations, large drill collar occupies, most of the fluid-filled borehole and divides the borehole fluid into two thin fluid columns （radius -27 mm）. Fine grids and large computational models are required to model the thin fluid region between the tool and the formation. As a result, small time step and more iterations are needed, which increases the cumulative numerical error. Furthermore, due to high impedance contrast between the drill collar and fluid in the borehole （the difference is 〉30 times）, the stability and efficiency of the perfectly matched layer （PML） scheme is critical to simulate complicated wave modes accurately. In this paper, we compared four different PML implementations in a staggered grid finite difference in time domain （FDTD） in the ALWD simulation, including field-splitting PML （SPML）, multiaxial PML（M- PML）, non-splitting PML （NPML）, and complex frequency-shifted PML （CFS-PML）. The comparison indicated that NPML and CFS-PML can absorb the guided wave reflection from the computational boundaries more efficiently than SPML and M-PML. For large simulation time, SPML, M-PML, and NPML are numerically unstable. However, the stability of M-PML can be improved further to some extent. Based on the analysis, we proposed that the CFS-PML method is used in FDTD to eliminate the numerical instability and to improve the efficiency of absorption in the PML layers for LWD modeling. The optimal values of CFS-PML parameters in the LWD simulation were investigated based on thousands of 3D simulations. For typical LWD cases, the best maximum value of the quadratic damping profile was obtained using one do. The optimal parameter space for the maximum value of the linear frequency-shifted factor （a0） and the scaling factor （β0） depended on the thickness of the PML layer. For typical formations, if the PML thickness is 10 grid points, the global error can be reduced to 〈1% using the optimal PML parameters, and the error will decrease as the PML thickness increases.     

Intermediate-term Predictions of Earthquakes in Italy: Algorithm M8

—Large earthquakes in Italy are preceded by a specific seismic activation which could be diagnosed by a reproducible intermediate-term earthquake prediction method—a modification for lower seismic rate areas of the algorithm, known as M8 (Keilis-Borok and Kossobokov, 1990). Use has been made of the PFG-ING catalog of earthquakes, compiled on a regular basis, to determine areas and times of increased probability for occurrences of M≥ 6 earthquakes. In retroactive simulation of forward prediction, for the period 1972–1995, both the 1976 Friuli, M = 6.1 and the 1980 Irpinia, M = 6.5 earthquakes are predicted. In the experiment where priority magnitude scale is used, the times of increased probability for a strong earthquake to occur (TIPs) occupy less than a quarter of the total magnitude-space-time domain, and are rather stable with respect to positioning of circles of investiga tion. Successful stability tests have been made considering a recently compiled catalog (CCI97) (Peresan et al., 1997). In combination with the CN algorithm results (Costa et al., 1996) the spatio-temporal uncertainty of the prediction could be reduced to 5%. The use of M8 for the forward prediction requires the computations to be repeated each half-year, using the updated catalog.     

A marching in space and time (MAST) solver of the shallow water equations. Part I: The 1D model

A new approach is presented for the numerical solution of the complete 1D Saint-Venant equations. At each time step, the governing system of partial differential equations (PDEs) is split, using a fractional time step methodology, into a convective prediction system and a diffusive correction system. Convective prediction system is further split into a convective prediction and a convective correction system, according to a specified approximated potential. If a scalar exact potential of the flow field exists, correction vanishes and the solution of the convective correction system is the same solution of the prediction system. Both convective prediction and correction systems are shown to have at each x − t point a single characteristic line, and a corresponding eigenvalue equal to the local velocity. A marching in space and time (MAST) technique is used for the solution of the two systems. MAST solves a system of two ordinary differential equations (ODEs) in each computational cell, using for the time discretization a self-adjusting fraction of the original time step. The computational cells are ordered and solved according to the decreasing value of the potential in the convective prediction step and to the increasing value of the same potential in the convective correction step. The diffusive correction system is solved using an implicit scheme, that leads to the solution of a large linear system, with the same order of the cell number, but sparse, symmetric and well conditioned. The numerical model shows unconditional stability with regard of the Courant–Friedrichs–Levi (CFL) number, requires no special treatment of the source terms and a computational effort almost proportional to the cell number. Several tests have been carried out and results of the proposed scheme are in good agreement with analytical solutions, as well as with experimental data.