Advanced sensitivity calibration of the Los Angeles strong motion array

Results are presented of recent sensitivity calibration of 76 accelerographs (SMA-1) of the Los Angeles Strong Motion Array. These have pendulum-like transducers and optical recording system. One characteristic of their design is off-axis sensitivity, which is magnified by transducer misalignment. A new calibration procedure was applied, which considers off-axis sensitivity and measures the angles of misalignment (φ and ψ), as well as the incident angle of the light beam onto the film (θ₀). These are required (1) for accurate estimation of sensitivity, and (2) for proper instrument correction of recorded accelerograms which considers also cross-axis sensitivity and misalignment. These effects are important near large acceleration peaks (approaching and exceeding 1g), e.g. like the ones recorded near the source of the 1994 Northridge earthquake (M_L=6·4). This earthquake was recorded by 65 stations of the Los Angeles Strong Motion Array, at epicentral distances from 2 to 85 km. Histograms showing distribution of the misalignment angles, light beam incidence angle θ₀ (for unloaded position) and the transducer sensitivities are presented. These indicate that the misalignment angles are typically 1–1·5°, but may also be 3–4°. Angle θ₀ (usually neglected), is mostly between ±8°, but may reach ±12°. Assuming θ₀=0 leads to systematically smaller values of the measured sensitivity (e.g. by ∼3% for θ₀=8° and ∼4% for θ₀=12°). Comparison of the newly measured sensitivities with those measured prior to installation (in 1979/1980), s^old, shows that, in general, the new values are systematically smaller. The difference is typically within 5 per cent, but in some cases is as large as 10 per cent. Other principal sources of the observed differences and their mechanisms are discussed. Those include long-term changes in the transducers (e.g. change of stiffness, reflected in changes of the natural frequency) and differences in the calibration procedure (e.g. errors associated with manual reading film records with tilt test data, and with transducer and instrument housing misalignment). The presented results may be considered typical of similar strong motion arrays worldwide. © 1998 John Wiley & Sons, Ltd.     

A note on the useable dynamic range of accelerographs recording translation

Since the late 1970s, the dynamic range and resolution of strong motion digital recorders have leaped from 65 to 135 dB, opening new possibilities for advanced data processing and interpretation. One of these new possibilities is the calculation of permanent displacement of the ground or of structures, associated with faulting or with non-linear response. Proposals on how permanent displacements could be recovered from recorded strong motion have been published since 1976. The analysis in this paper concludes that permanent displacements of the ground and of structures in the near-field can be calculated provided all six components of strong motion (three translations and three rotations) have been recorded, and the records are corrected for transducer rotation, misalignment and cross-axis sensitivity.     

An approximation method for solving the inverse MTS problem with the use of neural networks

The work develops the approximation approach to solving the inverse MTS problem with the use of neural networks. The inverse problem is considered in model classes of parametrized geoelectric structures, whose electric conductivity is controlled by a few hundreds of macroparameters (N ∼ 300). An approximate inverse operator of the problem is constructed for each model class as a neural network, whose coefficients are determined in the process of training on a representative sample of standard examples of forward problem solutions. The problem of determination of the model class of geolectric structures corresponding to the presented input MT data is solved with the use of the neural network classifier constructed for the available set of model classes of structures. Regularizing factors and errors of the neural network method are analyzed. The operation of the algorithm is illustrated by examples of the 2-D inversion of synthetic MT data.     

Inversion of electromagnetic data: An overview of new techniques

This paper explores some of the newer techniques for acquiring and inverting electromagnetic data. Attention is confined primarily to the 2d magnetotelluric (MT) problem but the inverse methods are applicable to all areas of EM induction. The basis of the EMAP technique of Bostick is presented along with examples to illustrate the efficacy of that method in structural imaging and in overcoming the deleterious effects of near-surface distortions of the electric field. Reflectivity imaging methods and the application of seismic migration techniques to EM problems are also explored as imaging tools. Two new approaches to the solution of the inverse problem are presented. The AIM (Approximate Inverse Mapping) inversion of Oldenburg and Ellis uses a new way to estimate a perturbation in an iterative solution which does not involve linearization of the equations. The RRI (Rapid Relaxation Inverse) of Smith and Booker shows how approximate Fréchet derivatives and sequences of 1d inversions can be used to develop a practical inversion algorithm. The overview is structured to provide insight about the latest inversion techniques and also to touch upon most areas of the inverse problem that must be considered to carry out a practical inversion. These include model parameterization, methods of calculating first order sensitivities, and methods for setting up a linearized inversion.     

A multiscale method for subsurface inverse modeling: Single-phase transient flow

High-resolution geologic models that incorporate observed state data are expected to effectively enhance the reliability of reservoir performance prediction. One of the major challenges faced is how to solve the large-scale inverse modeling problem, i.e., to infer high-resolution models from the given observations of state variables that are related to the model parameters according to some known physical rules, e.g., the flow and transport partial differential equations. There are typically two difficulties, one is the high-dimensional problem and the other is the inverse problem. A multiscale inverse method is presented in this work to attack these problems with the aid of a gradient-based optimization algorithm. In this method, the model responses (i.e., the simulated state data) can be efficiently computed from the high-resolution model using the multiscale finite-volume method. The mismatch between the observations and the multiscale solutions is then used to define a proper objective function, and the fine-scale sensitivity coefficients (i.e., the derivatives of the objective function with respect to each node’s attribute) are computed by a multiscale adjoint method for subsequent optimization. The difficult high-dimensional optimization problem is reduced to a one-dimensional one using the gradient-based gradual deformation method. A synthetic single-phase transient flow example problem is employed to illustrate the proposed method. Results demonstrate that the multiscale framework presented is not only computationally efficient but also can generate geologically consistent models. By preserving spatial structure for inverse modeling, the method presented overcomes the artifacts introduced by the multiscale simulation and may enhance the prediction ability of the inverse-conditional realizations generated.     

Inversion of dynamic production data for permeability: fast streamline-based computation of sensitivity coefficients of fractional flow rate

Generation of permeability field in a reservoir model that matchs historical dynamic production data requires an inverse calculation. A gradient method is typically used to solve the inverse minimization problem and requires sensitivity coefficients of reservoir responses, e.g. fractional flow rate or pressure, with respect to the change in the permeability. This paper presents a novel semi-analytical streamline-based method for computing such sensitivity coefficients under the framework of two-phase (oil-water) flow conditions. This method is shown to be significantly faster and generate permeability fields with lower objective function than the traditional perturbation method. The method decomposes the multiple-dimensional full flow problem into multiple 1D problems along streamlines. The sensitivity of fractional flow rate at the production well is directly related to the sensitivity of time-of-flight (TOF) along each individual streamline and the sensitivity of pressure at grid cells along the streamline. The sensitivity of TOF of a streamline can be obtained analytically. The sensitivity of pressure is obtained as part of a fast single phase flow simulation. The proposed method is implemented in a geostatistically based inverse technique, called the sequential self-calibration (SSC) method. Results for fractional flow rate sensitivities are presented and compared with the traditional perturbation method. This new method can be easily extended to compute sensitivity coefficients of saturation (concentration) data.     

声波方程逆散射反演的近似方法

我们在文献[1]里研究了介质参考波速沿某个方向线性变化时的三维声散射理论,导出了声波方程逆散射反演问题解的解析表达式.考虑到应用时的实际条件,本文根据上述反演方法导出2.5维模型的声波方程逆散射反演的波速扰动计算公式,给出该方法在“高频”近似条件下的波速扰动反演计算公式,从而使我们提出的“参考波速线性变化时的声波方程逆散射反演”理论更接近实际应用条件.本文给出的这些反演公式仍然具有原方法的优点,即不但可以使Born近似的假定在大多数情况下能得以满足,而且可以利用快速Fourier变换来快速实现介质波速扰动的反演成象.     

声波方程逆散射反演的近似方法

我们在文献[1]里研究了介质参考波速沿某个方向线性变化时的三维声散射理论,导出了声波方程逆散射反演问题解的解析表达式.考虑到应用时的实际条件,本文根据上述反演方法导出2.5维模型的声波方程逆散射反演的波速扰动计算公式,给出该方法在“高频”近似条件下的波速扰动反演计算公式,从而使我们提出的“参考波速线性变化时的声波方程逆散射反演”理论更接近实际应用条件.本文给出的这些反演公式仍然具有原方法的优点,即不但可以使Born近似的假定在大多数情况下能得以满足,而且可以利用快速Fourier变换来快速实现介质波速扰动的反演成象.     

用稳定高效的反Q滤波技术提高地震资料分辨率

地震波在地下传播时受到衰减影响,衰减会导致地震波场高频能量的损失和相位畸变.反Q滤波可补偿大地衰减效应.已有的反Q滤波方法存在下列不足:频率域的算法由于算子长度较长,所以计算效率较低;时间域的算法,或者对地震记录上到时较晚的同相轴进行了过度的补偿,或者为防止过度补偿后来的振幅而在最大增益处进行限制,导致振幅的多解性,而且还会影响滤波器的相位效应.本文给出一种通过直接求解时间域的Q模型方程来进行反Q滤波的算法.由于采用带状矩阵解算器,所以具有较高的计算效率,理论数据和实际地震资料的试算结果证明,本方法对地震波的吸收衰减进行了出色的补偿,提高了地震资料的分辨率.     

2D data modelling by electrical resistivity tomography for complex subsurface geology

A new tool for two‐dimensional apparent‐resistivity data modelling and inversion is presented. The study is developed according to the idea that the best way to deal with ill‐posedness of geoelectrical inverse problems lies in constructing algorithms which allow a flexible control of the physical and mathematical elements involved in the resolution. The forward problem is solved through a finite‐difference algorithm, whose main features are a versatile user‐defined discretization of the domain and a new approach to the solution of the inverse Fourier transform. The inversion procedure is based on an iterative smoothness‐constrained least‐squares algorithm. As mentioned, the code is constructed to ensure flexibility in resolution. This is first achieved by starting the inversion from an arbitrarily defined model. In our approach, a Jacobian matrix is calculated at each iteration, using a generalization of Cohn's network sensitivity theorem. Another versatile feature is the issue of introducing a priori information about the solution. Regions of the domain can be constrained to vary between two limits (the lower and upper bounds) by using inequality constraints. A second possibility is to include the starting model in the objective function used to determine an improved estimate of the unknown parameters and to constrain the solution to the above model. Furthermore, the possibility either of defining a discretization of the domain that exactly fits the underground structures or of refining the mesh of the grid certainly leads to more accurate solutions. Control on the mathematical elements in the inversion algorithm is also allowed. The smoothness matrix can be modified in order to penalize roughness in any one direction. An empirical way of assigning the regularization parameter (damping) is defined, but the user can also decide to assign it manually at each iteration. An appropriate tool was constructed with the purpose of handling the inversion results, for example to correct reconstructed models and to check the effects of such changes on the calculated apparent resistivity. Tests on synthetic and real data, in particular in handling indeterminate cases, show that the flexible approach is a good way to build a detailed picture of the prospected area.     

Robust inversion of vertical electrical sounding data using a multiple reweighted least-squares method

The root cause of the instability problem of the least-squares (LS) solution of the resistivity inverse problem is the ill-conditioning of the sensitivity matrix. To circumvent this problem a new LS approach has been investigated in this paper. At each iteration, the sensitivity matrix is weighted in multiple ways generating a set of systems of linear equations. By solving each system, several candidate models are obtained. As a consequence, the space of models is explored in a more extensive and effective way resulting in a more robust and stable LS approach to solving the resistivity inverse problem. This new approach is called the multiple reweighted LS method (MRLS). The problems encountered when using the L ₁- or L ₂-norm are discussed and the advantages of working with the MRLS method are highlighted. A five-layer earth model which generates an ill-conditioned matrix due to equivalence is used to generate a synthetic data set for the Schlumberger configuration. The data are randomly corrupted by noise and then inverted by using L ₂, L ₁ and the MRLS algorithm. The stabilized solutions, even though blurred, could only be obtained by using a heavy ridge regression parameter in L ₂- and L ₁-norms. On the other hand, the MRLS solution is stable without regression factors and is superior and clearer. For a better appraisal the same initial model was used in all cases. The MRLS algorithm is also demonstrated for a field data set: a stable solution is obtained.     

COMPARISON OF FIVE LEAST-SQUARES INVERSION TECHNIQUES IN RESISTIVITY SOUNDING*

A brief history of the development of the inverse problem in resistivity sounding is presented with the development of the equations governing the least-squares inverse. Five algorithms for finding the minimum of the least-square problem are described and their speed of convergence is compared on data from two planar earth models. Of the five algorithms studied, the ridge-regression algorithm required the fewest numbers of forward problem evaluations to reach a desired minimum. Solution space statistics, including (1) parameter-standard errors, (2) parameter correlation coefficients, (3) model parameter eigenvectors, and (4) data eigenvectors are discussed. The type of weighting applied to the data affects these statistical parameters. Weighting the data by taking log₁₀ of the observed and calculated values is comparable to weighting by the inverse of a constant data error. The most reliable parameter standard errors are obtained by weighting by the inverse of observed data errors. All other solution statistics, such as dataparameter eigenvector pairs, have more physical significance when inverse data error weighting is used.     

1D and 2D Cole-Cole-inversion of time-domain induced-polarization data

A new method for the 2D inversion of induced polarization (IP) data in the time domain has been developed. The entire IP transients were observed and inverted into 2D Cole-Cole earth models, including resistivity, chargeability, relaxation time and the frequency constant. Firstly, a modified 1D time-domain electromagnetic algorithm was used to calculate the response of a layered polarizable ground. The transient signals were then inverted using the Marquardt method to derive the Cole-Cole parameters of each layer. However, model calculations showed that the EM effects could be neglected for the time range (>1 ms) and for the transmitter–receiver distances (<50 m) used in this study. Therefore, the induction effects were not considered for the solution of the 2D inverse problem and a DC solution was applied. An approximative forward algorithm was introduced in order to calculate the IP transients directly in the time domain and in order to speed up the inverse procedure. The approximation is highly accurate, and this is demonstrated by comparing the approximations with their exact solutions up to 3D. The inverse algorithm presented consists of two steps. The transient voltages of an array data set were inverted separately into a two-dimensional resistivity model for each time channel. The time-dependent resistivity of each cell was then interpreted as the response of a homogeneous half-space. In the 2D inversion algorithm, a 3D DC algorithm was used as a forward operator. The method only requires a standard 2D DC inversion and a homogenous half-space Cole-Cole inversion. The developed algorithm has been successfully applied to synthetic data sets and to a field data set obtained from a waste site situated close to Düren in Germany.     

An evaluation of a class of practical optimization techniques for structural dynamics applications

This paper evaluates a class of practical optimization techniques for parameter identification of realistic structural dynamic systems. The techniques involve quasi-Newton methods together with an efficient procedure for estimating complicated error functions. The optimization procedures are verified through their application to several representative examples, including finite-element models of realistic structural systems. Extensive numerical and graphical results demonstrate the effects of various optimization algorithm parameters on the rate of convergence of the objective function, the parameter vector error norm and the gradient norm. Guidelines are presented as an aid for addressing several significant issues in the practical application of structural dynamics optimization procedures, such as sensitivity problems, uniqueness, initial value definition for the parameter vector, convergence rates, constraints, the effect of alternative cost function definitions, accuracy of alternative gradient evaluation procedures, alternative procedures for estimating the inverse of the Hessian matrix and the use of a quadratic approximation of the objective function.     

Simultaneous determination of aquifer parameters and zone structures with fuzzy c-means clustering and meta-heuristic harmony search algorithm

This study proposes an inverse solution algorithm through which both the aquifer parameters and the zone structure of these parameters can be determined based on a given set of observations on piezometric heads. In the zone structure identification problem fuzzy c-means (FCM) clustering method is used. The association of the zone structure with the transmissivity distribution is accomplished through an optimization model. The meta-heuristic harmony search (HS) algorithm, which is conceptualized using the musical process of searching for a perfect state of harmony, is used as an optimization technique. The optimum parameter zone structure is identified based on three criteria which are the residual error, parameter uncertainty, and structure discrimination. A numerical example given in the literature is solved to demonstrate the performance of the proposed algorithm. Also, a sensitivity analysis is performed to test the performance of the HS algorithm for different sets of solution parameters. Results indicate that the proposed solution algorithm is an effective way in the simultaneous identification of aquifer parameters and their corresponding zone structures.