The fields from a finite electrical dipole—A new computational approach

Geophysics ◽  
1994 ◽  
Vol 59 (6) ◽  
pp. 864-880 ◽  
Author(s):  
Kurt I. Sørensen ◽  
Niels B. Christensen
Keyword(s):  
Numerical Integration ◽  
Computation Time ◽  
Hankel Transform ◽  
Input Function ◽  
Earth Model ◽  
Model Parameters ◽  
Electrical Dipole ◽  
Hankel Transforms ◽  
Electromagnetic Methods ◽  
Order Of Magnitude

Controlled‐source, frequency‐domain, and time‐domain electromagnetic methods require accurate, fast, and reliable methods of computing the electric and magnetic fields from the source configurations used. Except for small magnetic dipole sources, all electric and magnetic sources are composed of lengths of straight wire, which may be grounded. If the source‐receiver separation is large enough, the composite electrical dipoles may be considered to be infinitely small, and in a 1-D earth model the fields are expressed as Hankel transforms of an input function, which depends only on the model parameters. The Hankel transforms can be evaluated using the digital filter theory of fast Hankel transforms. However, the approximation of the infinitely small dipole is not always valid, and fields from a finite electrical dipole must be calculated. Traditionally, this is done by numerical integration of the fields from an infinitesimal dipole, thus increasing computation time considerably. The fields from the finite electrical dipole are expressed as Hankel transforms and as integrals of Hankel transforms. The theory of fast Hankel transforms is extended to include integrals of Hankel transforms, and a method is devised for calculating the filter coefficients. Unlike the fast Hankel transform, the computation involved in the integrated Hankel transforms is not a true convolution, and so a set of filter coefficients must be calculated for each source‐receiver configuration. Furthermore, the method is extended to include the calculation of potential differences where one more integration is involved, which is what is actually measured in the field. The computation of filter coefficients is very fast, and for standard configurations, the coefficients need be computed only once. The method is as fast, accurate, and reliable as the fast Hankel transforms method, and is up to an order of magnitude faster than the usual numerical integration.

Numerical integration of related Hankel transforms of orders 0 and 1 by adaptive digital filtering

Geophysics ◽  
1979 ◽  
Vol 44 (7) ◽  
pp. 1287-1305 ◽  
Author(s):  
Walter L. Anderson
Keyword(s):  
Large Range ◽  
Digital Filtering ◽  
Hankel Transform ◽  
Kernel Functions ◽  
General Nature ◽  
Earth Model ◽  
Hankel Transforms ◽  
Quadrature Methods ◽  
Test Driver ◽  
Different Order

A linear digital filtering algorithm is presented for rapid and accurate numerical evaluation of Hankel transform integrals of orders 0 and 1 containing related complex kernel functions. The kernel for Hankel transforms is defined as the non‐Bessel function factor of the integrand. Related transforms are defined as transforms, of either order 0 or 1, whose kernel functions are related to one another by simple algebraic relationships. Previously saved kernel evaluations are used in the algorithm to obtain rapidly either order transform following an initial convolution operation. Each order filter is designed with identical abscissas over a large range so that an adaptive convolution procedure can be applied to a large class of kernels. Different order Hankel transforms with related kernels are often found in electromagnetic (EM) applications. Because of the general nature of this algorithm, the need to design new filters should not be necessary for most applications. Accuracy of the filters is comparable to that of single‐precision numerical quadrature methods, provided well‐behaved kernels and moderate values of the transform argument are used. Filtering errors of less than 0.005 percent are demonstrated numerically using known analytical Hankel transform pairs. The digital filter accuracy is also illustrated by comparison with other published filters for computing the apparent resistivity for a Schlumberger array over a horizontally layered earth model. The algorithm is written in Fortran IV and is listed in the Appendix along with a test driver program. Detailed comments are included to define sufficiently all calling parameter requirements.

Investigations of Tail and Defect States in a-Si0.6Ge0.4:H Alloys by PDS and ESR - Implications for the Interaction of Weak and Dangling Bonds

1995 ◽  
Vol 377 ◽  
Author(s):  
Tilo P. Drüsedau ◽  
Andreas N. Panckow ◽  
Bernd Schröder
Keyword(s):  
Silicon Germanium ◽  
Defect Density ◽  
State Density ◽  
Model Parameters ◽  
Defect States ◽  
Conversion Model ◽  
Order Of Magnitude ◽  
Underlying Mechanisms ◽  
Amorphous Silicon Germanium ◽  
Excess Absorption

ABSTRACTInvestigations on the gap state density were performed on a variety of samples of hydrogenated amorphous silicon germanium alloys (Ge fraction around 40 at%) containing different amounts of hydrogen. From subgap absorption measurements the values of the “integrated excess absorption” and the “defect absorption” were determined. Using a calibration constant, which is well established for the determination of the defect density from the integrated excess absorption of a-Si:H and a-Ge:H, it was found that the defect density is underestimated by nearly one order of magnitude. The underlying mechanisms for this discrepancy are discussed. The calibration constants for the present alloys are determined to 8.3×1016 eV−1 cnr2 and 1.7×1016 cm−2 for the excess and defect absorption, respectively. The defect density of the films was found to depend on the Urbach energy according to the law derived from Stutzmann's dangling bond - weak bond conversion model for a-Si:H. However, the model parameters - the density of states at the onset of the exponential tails N*=27×1020 eV−1 cm−3 and the position of the demarcation energy Edb-E*=0.1 eV are considerably smaller than in a-Si:H.

scholarly journals A Relation between Laplace and Hankel Transforms

1962 ◽  
Vol 5 (3) ◽  
pp. 114-115 ◽  
Author(s):  
B. R. Bhonsle
Keyword(s):  
Laplace Transform ◽  
Hankel Transform ◽  
Laplace And Hankel Transforms ◽  
Hankel Transforms ◽  
The Laplace Transform ◽  
Image Position

The Laplace transform of a function f(t) ∈ L(0, ∞) is defined by the equationand its Hankel transform of order v is defined by the equationThe object of this note is to obtain a relation between the Laplace transform of tμf(t) and the Hankel transform of f(t), when ℛ(μ) > − 1. The result is stated in the form of a theorem which is then illustrated by an example.

Development of an Inverse Plume Model for Mass Eruption Rate in Unsteady Conditions

2021 ◽  
Author(s):  
Stephen A Solovitz
Keyword(s):  
Laboratory Data ◽  
Model Performance ◽  
Volcanic Eruptions ◽  
Reynolds Numbers ◽  
Model Parameters ◽  
Plume Height ◽  
Eruption Rate ◽  
Mass Eruption Rate ◽  
Downstream Effects ◽  
Order Of Magnitude

Abstract Following volcanic eruptions, forecasters need accurate estimates of mass eruption rate (MER) to appropriately predict the downstream effects. Most analyses use simple correlations or models based on large eruptions at steady conditions, even though many volcanoes feature significant unsteadiness. To address this, a superposition model is developed based on a technique used for spray injection applications, which predicts plume height as a function of the time-varying exit velocity. This model can be inverted, providing estimates of MER using field observations of a plume. The model parameters are optimized using laboratory data for plumes with physically-relevant exit profiles and Reynolds numbers, resulting in predictions that agree to within 10% of measured exit velocities. The model performance is examined using a historic eruption from Stromboli with well-documented unsteadiness, again providing MER estimates of the correct order of magnitude. This method can provide a rapid alternative for real-time forecasting of small, unsteady eruptions.

Comparison of spectral methods for the evaluation of Stokes integral  

2021 ◽  
Author(s):  
Avadh Bihari Narayan ◽  
Ashutosh Tiwari ◽  
Govind Sharma ◽  
Balaji Devaraju ◽  
Onkar Dikshit
Keyword(s):  
Boundary Value Problems ◽  
Numerical Integration ◽  
Spectral Methods ◽  
Boundary Value ◽  
Fundamental Equation ◽  
Computation Time ◽  
Spherical Approximation ◽  
Integration Technique ◽  
Gravity Measurements ◽  
Stokes Integral

<p>The spherical approximation of the fundamental equation of geodesy defines the boundary value problems. Stokes’s integral provides the solution of boundary value problems that enables the computation of geoid from the properly reduced gravity measurements to the geoid. The stokes integral can be evaluated by brute-force numerical integration, spectral methods, and least-squares collocation. There is a trade-off between computation time and accuracy when we chose numerical integration technique or any spectral method. This research will compare time complexity and the accuracy of different spectral methods (1D-FFT, 2D-FFT, Multi-band FFT) and numerical integration technique for the region in the lower Himalaya, around Nainital, Uttarakhand, India. </p>

scholarly journals A Bayesian ensemble data assimilation to constrain model parameters and land-use carbon emissions

2018 ◽  
Vol 15 (9) ◽  
pp. 2909-2930 ◽  
Author(s):  
Sebastian Lienert ◽  
Fortunat Joos
Keyword(s):  
Land Use ◽  
Carbon Emissions ◽  
Latin Hypercube Sampling ◽  
Historical Period ◽  
Model Parameters ◽  
Dynamic Global Vegetation Model ◽  
Order Of Magnitude ◽  
The Difference ◽  
Wood Harvest ◽  
Global Vegetation

Abstract. A dynamic global vegetation model (DGVM) is applied in a probabilistic framework and benchmarking system to constrain uncertain model parameters by observations and to quantify carbon emissions from land-use and land-cover change (LULCC). Processes featured in DGVMs include parameters which are prone to substantial uncertainty. To cope with these uncertainties Latin hypercube sampling (LHS) is used to create a 1000-member perturbed parameter ensemble, which is then evaluated with a diverse set of global and spatiotemporally resolved observational constraints. We discuss the performance of the constrained ensemble and use it to formulate a new best-guess version of the model (LPX-Bern v1.4). The observationally constrained ensemble is used to investigate historical emissions due to LULCC (ELUC) and their sensitivity to model parametrization. We find a global ELUC estimate of 158 (108, 211) PgC (median and 90 % confidence interval) between 1800 and 2016. We compare ELUC to other estimates both globally and regionally. Spatial patterns are investigated and estimates of ELUC of the 10 countries with the largest contribution to the flux over the historical period are reported. We consider model versions with and without additional land-use processes (shifting cultivation and wood harvest) and find that the difference in global ELUC is on the same order of magnitude as parameter-induced uncertainty and in some cases could potentially even be offset with appropriate parameter choice.

Time Effects on Settlement of Rigid Pile Composite Foundation: Simplified Models

2018 ◽  
Vol 15 (07) ◽  
pp. 1850066 ◽  
Author(s):  
Meijuan Xu ◽  
Pengpeng Ni ◽  
Guoxiong Mei ◽  
Yanlin Zhao
Keyword(s):  
Numerical Integration ◽  
Half Space ◽  
Unsaturated Soils ◽  
Pressure Coefficient ◽  
Closed Form Solution ◽  
Elastic Half Space ◽  
Composite Foundation ◽  
Model Parameters ◽  
Time Effects ◽  
Rigid Pile

The behavior of pile composite foundation is studied using the flexibility method. During the analysis, determination of the flexibility matrix (settlement) is critical. However, conventional methods of Winkler and elastic half-space foundation models are incapable of considering the time effects of soil consolidation and creep. The foundation model of Zaretsky and Tsytovich [1965] can be used to evaluate settlement for unsaturated soils, but the complexity of numerical integration over an arbitrary loading area hinders its application. In this paper, a novel scheme is proposed for numerical integration by rotating the loading surface using the equiareal transformation technique. Therefore, a simplified closed-form solution is developed to calculate time dependent settlement for foundation soils. The efficacy of the proposed technique is demonstrated using illustrative examples of an elastic half-space, a rigid raft foundation without piles, and rigid pile composite foundations with multiple piles under surface loading. Furthermore, parametric study is conducted to evaluate the sensitivity of model parameters. The permeability [Formula: see text] and Poisson’s ratio [Formula: see text] are found to be important, whereas pore pressure coefficient [Formula: see text] and degree of saturation [Formula: see text] are less significant in the calculation.

scholarly journals Stable Numerical Evaluation of Finite Hankel Transforms and Their Application

2014 ◽  
Vol 2014 ◽  
pp. 1-11 ◽  
Author(s):  
Manoj P. Tripathi ◽  
B. P. Singh ◽  
Om P. Singh
Keyword(s):  
Stability Analysis ◽  
Heat Equation ◽  
Numerical Experiments ◽  
Numerical Evaluation ◽  
Radiation Condition ◽  
Hankel Transform ◽  
Basis Functions ◽  
Infinite Cylinder ◽  
Stable Algorithm ◽  
Hankel Transforms

A new stable algorithm, based on hat functions for numerical evaluation of Hankel transform of order ν>-1, is proposed in this paper. The hat basis functions are used as a basis to expand a part of the integrand, rf(r), appearing in the Hankel transform integral. This leads to a very simple, efficient, and stable algorithm for the numerical evaluation of Hankel transform. The novelty of our paper is that we give error and stability analysis of the algorithm and corroborate our theoretical findings by various numerical experiments. Finally, an application of the proposed algorithm is given for solving the heat equation in an infinite cylinder with a radiation condition.

scholarly journals An Efficient Implementation of the Finite-volume Method For the Solution of Radiation Transport in Circuit Breakers

2017 ◽  
Vol 4 (2) ◽  
pp. 177-181
Author(s):  
A. Mazaheri ◽  
J. Y. Trépanier ◽  
R. Camarero ◽  
P. Robin-Jouan
Keyword(s):  
Finite Volume ◽  
Radiation Transport ◽  
Computation Time ◽  
Discrete Ordinate Method ◽  
Discrete Ordinate ◽  
Circuit Breakers ◽  
Explicit Approach ◽  
Order Of Magnitude ◽  
Implicit Approach ◽  
Volume Method

In this paper, we propose to revisit the method to solve the radiation transport equation in circuit breakers to reduce the computation time. It is based on an explicit approach using a space marching algorithm. The method can further be accelerated using a Cartesian grid and using the axisymmetric assumption. Comparisons performed in terms of accuracy and efficiency between the P1 model, the implicit finite-volume discrete ordinate method and the space-marching finite-volume discrete ordinate method show that the explicit approach is more that an order of magnitude faster than the implicit approach, for the same accuracy.

Interpreting cardiac muscle force-length dynamics using a novel functional model

2004 ◽  
Vol 286 (4) ◽  
pp. H1535-H1545 ◽  
Author(s):  
Kenneth B. Campbell ◽  
Murali Chandra ◽  
Robert D. Kirkpatrick ◽  
Bryan K. Slinker ◽  
William C. Hunter
Keyword(s):  
Cardiac Muscle ◽  
Functional Model ◽  
Input Function ◽  
Model Parameters ◽  
Dynamic Force ◽  
Cross Bridges ◽  
Cardiac Muscle Fibers ◽  
The Time Domain ◽  
Validation Tests ◽  
Distortion Parameters

To describe the dynamics of constantly activated cardiac muscle, we propose that length affects force via both recruitment and distortion of myosin cross bridges. This hypothesis was quantitatively tested for descriptive and explanative validity. Skinned cardiac muscle fibers from animals expressing primarily α-myosin heavy chain (MHC) (mouse, rat) or β-MHC (rabbit, ferret) were activated with solutions from pCa 6.1 to 4.3. Activated fibers were subjected to small-amplitude length perturbations [Δ L( t)] rich in frequency content between 0.1 and 40 Hz. In descriptive validation tests, the model was fit to the ensuing force response [ΔF( t)] in the time domain. In fits to 118 records, the model successfully accounted for most of the measured variation in ΔF( t) ( R2 range, 0.997–0.736; median, 0.981). When some residual variations in ΔF( t) were not accounted for by the model (as at low activation), there was very little coherence (<0.5) between these residual force variations and the applied Δ L( t) input function, indicating that something other than Δ L( t) was causing the measured variation in ΔF( t). With one exception, model parameters were estimated with standard errors on the order of 1% or less. Thus parameters of the recruitment component of the model could be uniquely separated from parameters of the distortion component of the model and parameters estimated from any given fiber could be considered unique to that fiber. In explanative validation tests, we found that recruitment and distortion parameters were positively correlated with independent assessments of the physiological entity they were assumed to represent. The recruitment distortion model was judged to be valid from both descriptive and explanative perspectives and is, therefore, a useful construct for describing and explaining dynamic force-length relationships in constantly activated cardiac muscle.

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