Numerical Inversion Methods Research Articles

Bayesian-based estimation of acoustic surface impedance: Finite difference frequency domain approach.

Acoustic performance for an interior requires an accurate description of the boundary materials' surface acoustic impedance. Analytical methods may be applied to a small class of test geometries, but inverse numerical methods provide greater flexibility. The parameter estimation problem requires minimizing prediction vice observed acoustic field pressure. The Bayesian-network sampling approach presented here mitigates other methods' susceptibility to noise inherent to the experiment, model, and numerics. A geometry agnostic method is developed here and its parameter estimation performance is demonstrated for an air-backed micro-perforated panel in an impedance tube. Good agreement is found with predictions from the ISO standard two-microphone, impedance-tube method, and a theoretical model for the material. Data by-products exclusive to a Bayesian approach are analyzed to assess sensitivity of the method to nuisance parameters.

Using measured soil water contents to estimate evapotranspiration and root water uptake profiles – a comparative study

Abstract. Understanding the role of plants in soil water relations, and thus ecosystem functioning, requires information about root water uptake. We evaluated four different complex water balance methods to estimate sink term patterns and evapotranspiration directly from soil moisture measurements. We tested four methods. The first two take the difference between two measurement intervals as evapotranspiration, thus neglecting vertical flow. The third uses regression on the soil water content time series and differences between day and night to account for vertical flow. The fourth accounts for vertical flow using a numerical model and iteratively solves for the sink term. None of these methods requires any a priori information of root distribution parameters or evapotranspiration, which is an advantage compared to common root water uptake models. To test the methods, a synthetic experiment with numerical simulations for a grassland ecosystem was conducted. Additionally, the time series were perturbed to simulate common sensor errors, like those due to measurement precision and inaccurate sensor calibration. We tested each method for a range of measurement frequencies and applied performance criteria to evaluate the suitability of each method. In general, we show that methods accounting for vertical flow predict evapotranspiration and the sink term distribution more accurately than the simpler approaches. Under consideration of possible measurement uncertainties, the method based on regression and differentiating between day and night cycles leads to the best and most robust estimation of sink term patterns. It is thus an alternative to more complex inverse numerical methods. This study demonstrates that highly resolved (temporally and spatially) soil water content measurements may be used to estimate the sink term profiles when the appropriate approach is used.

Performance comparison of numerical inversion methods for Laplace and Hankel integral transforms in engineering problems

Different methods for the numerical evaluations of the inverse Laplace and inverse of joint Laplace–Hankel integral transforms are applied to solve a wide range of initial-boundary value problems often arising in engineering and applied mathematics. The aim of the paper is to present a performance comparison among different numerical methods when they are applied to transformed functions related to actual engineering problems found in the literature. Most of our selected test functions have been found in the solution of boundary value problems of applied mechanics such as those related to transient responses of isotropic and transversely isotropic half-space to concentrated impulse or those related to viscoelastic wave motion in layered media. These classes of test functions are frequently encountered in similar problems such as those in boundary element or boundary integral equations, theoretical seismology, soil–structure-interaction in time domain and so on. Therefore, their behavior with different numerical inversion algorithms could make a useful guide to a precise choice of more suitable inversion method to be used in similar problems. Some different methods are also investigated in detail and compared for the inversion of the joint Hankel–Laplace transforms, where more sophisticated integrand functions are encountered. It is shown that Durbin, Crump, D’Amore, Fixed-Talbot, Gaver–Whyn–Rho (GWR), and Direct Integration methods have excellent performance and produce good results when applied to the same problems. On the contrary, Gaver–Stehfest and Piessens methods furnish results not very reliable for almost all classes of transformed functions and they seem good only for “simple” transformed functions. Particularly the performance of GWR algorithm is very good even for transformed functions with infinite number of singularities, where the other methods fail. In addition, in case of double integral transforms, only the Fixed-Talbot, Durbin and Weeks methods are recommended.

On different numerical inverse Laplace methods for solute transport problems

Numerical inversion is required when Laplace transform cannot be inverted analytically by manipulating tabled formulas of special cases. However, the numerical inverse Laplace transform is generally an ill-posed problem, and there is no universal method which works well for all problems. In this study, we selected seven commonly used numerical inverse Laplace transform methods to evaluate their performance for dealing with solute transport in the subsurface under uniform or radial flow condition. Such seven methods included the Stehfest, the de Hoog, the Honig–Hirdes, the Talbot, the Weeks, the Simon and the Zakian methods. We specifically investigated the optimal free parameters of each method, including the number of terms used in the summation and the numerical tolerance. This study revealed that some commonly recommended values of the free parameters in previous studies did not work very well, especially for the advection-dominated problems. Instead, we recommended new values of the free parameters for some methods after testing their robustness. For the radial dispersion, the de Hoog, the Talbot, and the Simon methods worked very well, regardless of the dispersion-dominated or advection-dominated situations. The Weeks method can be used to solve the dispersion-dominated problems, but not the advection-dominated problems. The Stehfest, the Honig–Hirdes, and the Zakian methods were recommended for the dispersion-dominated problems. The Zakian method was efficient, while the de Hoog method was time-consuming under radial flow condition. Under the uniform flow condition, all the methods could present somewhat similar results when the free parameters were given proper values for dispersion-dominated problems; while only the Simon method, the Weeks method, and the de Hoog method worked well for advection-dominated problems.

Analytical solutions for flow of horizontal well in compressible, three-dimensional unconsolidated formations

A novel mathematical model for single-phase fluid flow from unconsolidated formations to a horizontal well with the consideration of stress-sensitive permeability is presented. The model assumes the formation permeability is an exponential function of the pore pressure. Using a perturbation technique, the model is solved for either constant pressure or constant flux or infinite lateral boundary conditions with closed top and bottom boundaries. Through Laplace transformation, finite Fourier transformation and numerical inversion methods, the solutions are obtained and the pressure response curves are analyzed. The agreement between the analytical solutions in this paper and the numerical results from commercial software (Saphir) is excellent, which manifests the accuracy of the results derived in this paper.

Structural Health and Load Monitoring with Material-embedded Sensor Networks and Self-organizing Multi-agent Systems

One of the major challenges in Structural Health Monitoring and load monitoring of mechanical structures is the derivation of meaningful information from sensor data. This work investigates a hybrid data processing approach for material-integrated SHM and LM systems by using self-organizing mobile multi-agent systems (MAS), with agent processing platforms scaled to microchip level which offer material-integrated real-time sensor systems, and inverse numerical methods providing the spatial resolved load information from a set of sensors embedded in the technical structure. Inverse numerical approaches usually require a large amount of computational power and storage resources, not suitable for resource constrained sensor node implementations. Instead, off-line computation is performed, with on-line sensor processing by the agent system.

Investigation on Transient Responses of a Piezoelectric Crack by using Durbin and Zhao Methods for Numerical Inversion of Laplace Transforms

ABSTRACTThis study applies the numerical inversion of Laplace transform methods to study the piezoelectric dynamic fracture problem, recalculating Chen and Karihaloo's [1] analysis on the transient response of a impermeable crack subjected to anti-plane mechanical and in-plane electric impacts. Three numerical methods were adopted for calculating the dynamic stress intensity factor: Durbin method, Zhao method 1, and Zhao method 2. The results obtained were more accurate than the results in Chen and Karihaloo's [1] study. Through the calculation, this study presents a better range of parameters for the above three methods, and compares the advantages and disadvantages of each method in detail.

Identification of natural gas fractured volcanic formation by using numerical inversion method

Abstract The difference between formation resistivity and tight surrounding rock resistivity is caused by reservoir fractures and fluid property in the pore space, when factors such as mudstone, pore and other conductive minerals are eliminated. Therefore, the use of volcanic lithology identification, the resistivity of gas-bearing reservoir, and tight surrounding layer with the same lithology are selected to determine the formation true resistivity formula for gas-bearing intervals, which eliminates the factors of reservoir factures and fluid property in the pore space through numerical inversion methods. Thus, the difference-ratio value of structural resistivity is defined as the ratio of difference between formation true resistivity and deep investigation laterolog resistivity to deep investigation laterolog resistivity. Such parameters reflect the influence of reservoir fractures and fluid property in the pore space for the reduced rate of resistivity, so it identifies fluid property of volcanic fractured formation. Finally, the identification plate of oil layer, gas layer, and water layer for volcanic reservoir is established. Therefore, the natural gas layers of volcanic fractured formation are effectively evaluated and identified in combination with other well logging curves, reservoir performance data, and gas surveying information.

Pressure Transient Analysis of Dual Fractal Reservoir

A dual fractal reservoir transient flow model was created by embedding a fracture system simulated by a tree-shaped fractal network into a matrix system simulated by fractal porous media. The dimensionless bottom hole pressure model was created using the Laplace transform and Stehfest numerical inversion methods. According to the model's solution, the bilogarithmic type curves of the dual fractal reservoirs are illustrated, and the influence of different fractal factors on pressure transient responses is discussed. This semianalytical model provides a practical and reliable method for empirical applications.

Analysing tension infiltrometer data from sloped surface using two-dimensional approximation

Tension infiltrometers allow water to infiltrate into the soils at various specified pressure heads. The resulting infiltration rates can then be analysed for soil hydraulic properties by either analytical or inverse numerical methods. Tension infiltrometers however are primarily designed to be deployed on horizontal land surfaces, and their applications have been studied widely using an inverse numerical tool HYDRUS-2D. However, natural landscapes are often nonhorizontal, and infiltration through tension infiltrometers on sloped surfaces is no longer an axisymmetrical two-dimensional (2D) process but a fully three-dimensional (3D) one. In addition, minimal research has examined the effect of simplifying the 3D problem to a 2D one on the hydraulic conductivity estimated using tension infiltrometer data from different land slopes of various soil types. Therefore, in this study, tension infiltrometer data on different slopes have been obtained from a catchment located at National University of Singapore. In addition, tension infiltrometer data of six soil types on different slopes and with different initial water content were simulated using HYDRUS-3D. Combining field measurements, forward and inverse modelling, the influence of applying a 2D approximation on hydraulic property estimations using tension infiltrometer data was examined. The results show that the estimations for soils with high infiltration rates are more sensitive to application of the 2D approximation. The maximum allowable slopes for employing 2D approximation on clayey and sandy soils are 25° and less than 3°, respectively. Furthermore, the maximum allowable slope decreases with decreasing initial water content. Copyright © 2012 John Wiley & Sons, Ltd.

Tomographic GPR imaging using a linear inversion algorithm informed by FDTD modelling: a numerical case study of buried utility pipes monitoring

ABSTRACTIn this paper, we present the results of an on‐going study into the development of a practical strategy for the interpretation and analysis of near‐surface GPR data in complex scenarios. In particular, we consider the problem of how the knowledge of the investigated scenario can be exploited to improve the diagnostic results using an approach based on the joint use of FDTD numerical modelling and linear tomographic inversion methods. The performance of the approach is evaluated using a simulated test‐case in which GPR data are collected in a complex utility‐pipe model. Prior knowledge of the investigation scenario is captured for the inversion using a three‐dimensional, full‐field FDTD modelling scheme to calculate the incident field and the Green’s functions, allowing the antenna geometry, the air‐ground interface and known subsurface a priori information to be accounted for. As input data to the inversion algorithm, we assume the raw GPR data preprocessed only by simple time‐gating, after Truncated Singular Value Decomposition (TSVD) resulting from the ‘informed linear’ model, are exploited to achieve a regularized solution of the problem. The results show that with just this basic assumption, the joint use of these two (forward and inverse) modelling techniques enhances tomographic imaging in very complex scenarios.

Study of heat-transfer characteristics on the fin of two-row plate finned-tube heat exchangers

The present study applies the experimental and numerical inverse methods to determine the average heat transfer coefficient h¯ and heat transfer coefficient under the isothermal situation h¯iso on a vertical square fin of the two-row plate finned-tube heat exchangers with four circular tubes for the staggered arrangement and various air velocities and fin spacings. These tubes may not have the same temperature. The experimental fin temperature measurements, ambient air temperature and air velocity are obtained from the present small wind tunnel. Due to the non-uniform distribution of the heat transfer coefficient, the square plate fin is divided into several sub-fin regions before performing the inverse scheme, and the average heat transfer coefficient in each sub-fin region is assumed to be unknown. The results show that the h¯ and h¯iso values increase with the air velocity and fin spacing for the staggered arrangement. In order to validate the reliability of the present results, the h¯iso value is compared with the existing correlations for the staggered arrangement.

Unified geophysical and geological 3D Earth models

Three-dimensional geological Earth models typically comprise wireframe surfaces of connected triangles that represent geological contacts. In contrast, Earth models used by most current 3D geophysical numerical modeling and inversion methods are built on rectilinear meshes. This is because the mathematics for computing data responses are simpler on rectilinear meshes. In such a model, the relevant physical properties are uniform within each brick-like cell but possibly different from one cell to the next, producing a pixellated representation of the Earth. In principle, arbitrary spatial variations can be represented if a sufficiently fine discretization is used. However, no matter how fine the discretization of the rectilinear mesh, such a mesh is always incompatible with geological models comprising wireframe surfaces. Also, because the computational resources required by 3D numerical modeling and inversion methods increase dramatically as the discretization of a model is refined, it is never really possible to achieve as fine a discretization as one would like. This exacerbates the mismatch between models that comprise wireframe surfaces and those built on rectilinear meshes. To address this incompatibility, we are using unstructured tetrahedral meshes to specify 3D geophysical Earth models. We hope that working with unstructured meshes will facilitate the construction of common Earth models consistent with both the geological and geophysical data available.

A Review on the Numerical Inversion Methods of Relative Permeability Curves

In general, relative permeability data can be obtained from laboratory coreflooding experiments. Such experimental data can be interpreted analytically or numerically. Compared to analytical methods, when the numerical inversion methods are applied to interpret the coreflooding experimental data, the reservoir performance obtained prior to and after breakthrough can be utilized comprehensively, the capillary effects and the heterogeneity of core samples can also be taken into account, so the estimated result is not only accurate but also complete. Moreover, the numerical inversion methods can be applied to large-scale reservoirs. This article introduces systematically the methodology of numerical inversion methods, and then reviews the present research status. Finally, several proposals of implicitly estimating relative permeability data are put forward from aspects of optimization algorithms’ properties, estimation of endpoint saturations and treatment scale by automatic history matching.

Determination of heat transfer coefficients by extrapolation and numerical inverse methods in squeeze casting of magnesium alloy AM60

In this work, a different wall-thickness 5-step (with thicknesses as 3, 5, 8, 12, 20 mm) casting mold was designed, and squeeze casting of magnesium alloy AM60 was performed under an applied pressure 30, 60 and 90 MPa in a hydraulic press. The casting–die interfacial heat transfer coefficients (IHTC) in the 5-step casting were determined based on thermal histories throughout the die and inside the casting which were recorded by fine type-K thermocouples. With measured temperatures, heat flux and IHTCs were evaluated using the polynomial curve fitting method and numerical inverse method. For numerical inverse method, a solution algorithm was developed based on the function specification method to solve the inverse heat conduction equations. The IHTCs curves for five steps versus time were displayed. As the applied pressures increased, the IHTC peak value of each step was increased accordingly. It can be observed that the peak IHTC value decreased as the step became thinner. Furthermore, the accuracy of these curves was analyzed by the direct modeling calculation. The results indicated that heat flux and IHTCs determined by the inverse method were more accurately than those from the extrapolated fitting method.

Mathematical Modeling of Bivariate Polymer Property Distributions Using 2D Probability Generating Functions, 1 – Numerical Inversion Methods

AbstractThis is the first of two papers presenting a new mathematical method for modeling bivariate distributions of polymer properties. It is based on the transformation of the infinite mass balances describing the evolution of a two‐dimensional distribution using 2D probability generating functions (pgf). A key step of this method is the inversion of the transforms. In this work, two numerical inversion methods of 2D pgfs are developed and carefully validated. The accuracy obtained with both methods was very satisfactory. The inversion formulas of both methods are simple and easy to implement. A simple copolymerization example is used to show the complete procedure from the derivation of the pgf balances to the recovery of the bivariate molecular weight distribution.magnified image

Interpreting complex, three‐dimensional, near‐surface GPR surveys: an integrated modelling and inversion approach

ABSTRACTWith the increasing computational power of modern personal computers, sophisticated modelling and inversion techniques are becoming popular tools for the interpretation of high‐resolution, fully three‐dimensional GPR surveys. In this paper, we present the latest results of ongoing practical research into the development of novel, integrated, finite‐difference time‐domain (FDTD) numerical modelling and linear tomographic inversion methods for the interpretation and analysis of near–surface, 3D GPR data. The proposed approach utilizes the Born approximation solution to the inverse‐scattering problem and a truncated singular value decomposition (TSVD) to create the final, inverted reconstructions. A three‐dimensional, full‐field, O(2,4) accurate FDTD modelling scheme is used to generate the numerical‐based Green’s functions and incident fields for the inversion. As such, accurate antenna sources (including the influence of shields) and near‐field air/ground interface effects are inherently included in the inversion formulation. The performance of this integrated method is evaluated via a simulated, 3D, forensic‐based, test‐case example (a 900 MHz survey over a clandestine human burial target) including coherent noise from near‐surface clutter. Although the example is simplistic, the results show that the scheme works well, despite some assumptions in the inversion methodology. As such, useful information can be gained on the true form, depth, location and spatial interrelationships of the buried features and, therefore, improved interpretations can be obtained in a three‐dimensional context.

Laboratory and field testing of an automated tension infiltrometer

Three identical newly developed automated tension infiltrometers ( Špongrová et al. , 2009 ) were tested in the laboratory and in the field. In the laboratory, the infiltrometers were used on a uniform sandy loam profile prepared in a large soil tank. The measured steady-state data were used to determine the hydraulic conductivity near saturation, K ( h ), at 8 tensions ranging from 1 to 13 cm using the analytical method proposed by Reynolds and Elrick (1991) . In addition, transient cumulative flow data were used to determine the soil hydraulic functions via inverse modelling using the numerical model HYDRUS-2D. Analysis of variance showed that the analytical and numerical methods did not differ from each other within the tension range used in this study. The tension infiltrometers were subsequently used in the field on a sandy loam soil to characterise five different tillage treatments (conventional plough (CP), shallow plough (SP), minimum tillage (MT), direct drill (DD), and no-treatment (NT)). The effect of wheel traffic was also evaluated by measuring the infiltration rates in the wheel marks. The results show that the MT and NT plots had a higher hydraulic conductivity near saturation than the tilled soils which did not differ significantly from each other. In addition wheel traffic leads to soil compaction and a significant reduction in the hydraulic conductivity close to saturation. The automated system tested in this study allowed the measurement of infiltration rates for 8 tensions in triplicate per day with no human intervention apart for refilling the reservoir, and provided transient and steady-state data that can be analysed using both analytical and numerical inversion methods.

Three-dimensional semi-analytical solution to groundwater flow in confined and unconfined wedge-shaped aquifers

The Laplace domain solutions have been obtained for three-dimensional groundwater flow to a well in confined and unconfined wedge-shaped aquifers. The solutions take into account partial penetration effects, instantaneous drainage or delayed yield, vertical anisotropy and the water table boundary condition. As a basis, the Laplace domain solutions for drawdown created by a point source in uniform, anisotropic confined and unconfined wedge-shaped aquifers are first derived. Then, by the principle of superposition the point source solutions are extended to the cases of partially and fully penetrating wells. Unlike the previous solution for the confined aquifer that contains improper integrals arising from the Hankel transform [Yeh HD, Chang YC. New analytical solutions for groundwater flow in wedge-shaped aquifers with various topographic boundary conditions. Adv Water Resour 2006;26:471–80], numerical evaluation of our solution is relatively easy using well known numerical Laplace inversion methods. The effects of wedge angle, pumping well location and observation point location on drawdown and the effects of partial penetration, screen location and delay index on the wedge boundary hydraulic gradient in unconfined aquifers have also been investigated. The results are presented in the form of dimensionless drawdown-time and boundary gradient-time type curves. The curves are useful for parameter identification, calculation of stream depletion rates and the assessment of water budgets in river basins.

Wideband leaky-mode resonance reflectors: Influence of grating profile and sublayers

We apply inverse numerical methods to design compact wideband reflectors in which a periodic silicon layer supports resonant leaky modes. Using particle swarm optimization to determine appropriate device thickness, period, and fill factors, we arrive at example reflector designs for both TE and TM polarized input light. As a properly configured grating profile provides added design freedom, we design reflectors with two and four subparts in the period. In TM polarization, a particular single-layer two-part reflector has 520 nm bandwidth whereas the four-part device reaches 600 nm bandwidth. In TE polarization, the corresponding numbers are 125 nm and 495 nm, respectively. We provide a qualitative explanation for the smaller TE-reflector bandwidth. We quantify the effects of deviation from the design parameters and compute the angular response of the elements. As the angle of incidence deviates from normal incidence, narrow transmission channels emerge in the response yielding a bandpass filter with low sidebands. The effects of adding a silica sublayer between a silicon substrate and the periodic silicon layer is investigated. It is found that a properly designed sublayer can extend the reflection bandwidth significantly.