A variable interval width quadrature technique based on Romberg's method

Miller, Edmund K

doi:10.1016/0021-9991(70)90063-x

Cited by 17 publications

(4 citation statements)

References 6 publications

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“…A strategy similar to one found useful for adaptive numerical quadrature, based on Romberg's method (Romberg's method) [43], might be adapted to the spectral-estimation problem. That adaptive procedure involves choosing a starting subinterval, over which five successive uniformly spaced samples of an integrand are computed.…”

Section: Adaptive Sampling Strategiesmentioning

confidence: 99%

Model-based parameter estimation in electromagnetics. II. Applications to EM observables

Miller

1998

IEEE Antennas Propag. Mag.

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6. Spectral-domain MBE'E 6.1 Antenna Applications he MBPE (model-based parameter estimation) procedure T described above has been tested for a variety of wave-equation phenomena, of which seFeral examples are included here. Plotted in Figure 2 are the input conductance and susceptance of a monopole, as a function of the antenna size in wavelengths [26]. These results were obtained by sampling a generating model (GM) (NEC) at 0.15 intervals in L//z, shown by the open crosses. The crosses 30 ! I I I" + NEC Samples 5 2 5 -----Linear Interpolationv E FM Result w20 -0 . . z 0 0 5 I 1 0 1 2 3 LengthWavelength Figure 2a. The input conductance of a monopole antenna, versus the length in wavelengths, as obtained from a series of overlapping rational-function-fitting models. This used d = n = 3 (the solid line), based on 20 generating-model samples spaced 0.15 apart in Llwavelength, which are shown as open crosses and joined by a dashed line [26]. A comparison of the fitting-model values with the generating-model samples at other frequencies reveal's a numerical agreement of 1% or better. -'Editor's note: Part I of this article appeared in the February issue of the Magazine; Part I11 will appear in the June issue. 0 F 2-3 Length/Wavelength Figure 2b. The input susceptance of the monopole antenna, as obtained by the method used for Figure 2a. are connected by the dashed lines, to show the result of straightline interpolation; the MBPE results are the solid line.The fitting model (FM) used a d = n = 3 rational function, which requires seven data points. The first fitting model, M I , was used to develop an estimate out to the fourth data sample, as shown by the solid line. The second model, M,, was "slid" up in frequency by adding the eighth data sample and dropping the first. It was used to plot the solid line between the fourth and fifth data samples. This process was continued until model M14 was reached, which completed the fitting-model plot from data points 16 to 20. This particular procedure is not unique, as other approaches might be used, but it produces results within a few per cent of the generating model, between the data samples actually used to quantify the fitting model. Extensions of this idea include comparing the fitting models for mutual consistency in their regions of overlap, to estimate the relative uncertainty of the fitting-model result, and to determine whether additional generatingmodel samples are needed.A similar use of MBPE for a more complicated problem, representing the input impedance of a log-periodic dipole array (LPDA) due to de Beer and Baker [27], is demonstrated in Figure 3. (Note that the rational-function fitting model can be applied equally well to admittance functions, as shown in Figure 2, or to

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Section: Adaptive Sampling Strategiesmentioning

confidence: 99%

Model-based parameter estimation in electromagnetics. II. Applications to EM observables

Miller

1998

IEEE Antennas Propag. Mag.

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“…The integrals to infinity were evaluated numerically with an adaptive Romberg quadrature routine [15]. Typically fewer than 20 integrand evaluations are needed since both the exponential function and the Hankel function in Gc decay exponentially on the deformed contours.…”

Section: The 2-d Integral Equation Modelmentioning

confidence: 99%

A physical optics model for scattering by irregular terrain at HF

Burke¹

1994

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“…The search is continued in the same manner from the endpoint of that segment whereon the magnitude of the integrand has a minimum value. Numerical evaluation of the integral along each segment is performed using variable width Romberg quadrature (Miller 1970). The process continues until the contribution along the last segment is low3 of the total sum.…”

Section: R E F E R E N C E S O L U T I O Nmentioning

confidence: 99%

Modes to replace transitional asymptotic ray fields in a vertically inhomogeneous earth model

Niver

Kamel

Felsen

1985

Geophysical Journal International

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Asymptotic ray theory (ART) fails in transition regions near critically reflected, bottom glancing or caustic-forming rays in a vertically inhomogeneous layered earth. These deficiencies are repaired here by replacing the transitional ray fields with guided modes plus truncation remainders. Exact ray-mode equivalences and their high-frequency asymptotic approximations are formulated, and their validity and efficiency are verified by numerical comparisons for SH motion in a two-layer earth model comprised of an inhomogeneous sediment above an homogeneous semi-infinite bedrock. While these considerations have granted valuable insight into the propagation characteristics of groups of mode fields and groups of ray fields, they have not, until recently, been placed on a rigorous basis to permit their use as a quantitative alternative. By this rigorous approach (for a summary, see Felsen 1984), a given finite spectral wavenumber (slowness) interval can be filled either with ray fields or with mode fields, with inclusion of remainder fields that fill possible spectral voids near the edges of the interval when, as is usually the case, the spectral spread of the one is not exactly the same as the spectral spread of the other. The previously referred to smearing out of the discrete mode index into a continuum is here embedded within a rigorous framework, as is the inclusion of truncation remainders that were neglected entirely in the earlier qualitative treatments.The bilateral exact ray-mode equivalence can be employed in several ways. It can quantify the truncation remainder when a ray series or a normal mode series is truncated at any element. The remainder (truncation error) can be expressed as an integral which may either be computed directly or reduced to modes plus an often negligible but always smaller remainder integral (for reference, see Felsen 1984). The integrals are defined over contours in the complex wavenumber plane whereon the integrand decays exponentially and therefore permits efficient asymptotic or numerical evaluation. The equivalence can also be employed to excise from a ray series a group of elements and replace these with modes (plus 'ray remainders'), or to excise from a mode series a group of elements and replace these with rays (plus 'mode remainders'). There results then a hybrid representation comprising uniquely determined successive groups of rays and groups of modes, with interspersed remainders. The replacements can be structured so as to remove troublesome elements and thereby render the calculation of the overall seismogram more efficient. This leaves a great deal of flexibility for the user in deciding on strategy and exercising various options.We have previously applied the hybrid formulation to construct a seismogram for SH motion in a simple configuration comprising a homogeneous sediment above a homogeneous bedrock, with the source located in the bedrock (Kamel & Felsen 1981). The results in this case clearly demonstrated the advantages of the hybrid approach. In the present paper,...

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A variable interval width quadrature technique based on Romberg's method

Cited by 17 publications

References 6 publications

Model-based parameter estimation in electromagnetics. II. Applications to EM observables

Model-based parameter estimation in electromagnetics. II. Applications to EM observables

A physical optics model for scattering by irregular terrain at HF

Modes to replace transitional asymptotic ray fields in a vertically inhomogeneous earth model

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