Electronic Resource
Oxford, UK
:
Blackwell Publishing Ltd
Geophysical prospecting
27 (1979), S. 0
ISSN:
1365-2478
Source:
Blackwell Publishing Journal Backfiles 1879-2005
Topics:
Geosciences
,
Physics
Notes:
Inspired by the linear filter method introduced by D. P. Ghosh in 1970 we have developed a general theory for numerical evaluation of integrals of the Hankel type:〈displayedItem type="mathematics" xml:id="mu1" numbered="no"〉〈mediaResource alt="image" href="urn:x-wiley:00168025:GPR876:GPR_876_mu1"/〉Replacing the usual sine interpolating function by sinsh (x) =a· sin (ρx)/sinh (aρx), where the smoothness parameter a is chosen to be “small”, we obtain explicit series expansions for the sinsh-response or filter function H*.If the input function f(λ exp (iω)) is known to be analytic in the region o 〈 λ 〈 ∞, |ω|≤ω0 of the complex plane, we can show that the absolute error on the output function is less than (K(ω0)/r) · exp (−ρω0/Δ), Δ being the logarthmic sampling distance.Due to the explicit expansions of H* the tails of the infinite summation 〈displayedItem type="mathematics" xml:id="mu2" numbered="no"〉〈mediaResource alt="image" href="urn:x-wiley:00168025:GPR876:GPR_876_mu2"/〉 ((m−n)Δ) can be handled analytically.Since the only restriction on the order is ν 〉 − 1, the Fourier transform is a special case of the theory, ν=± 1/2 giving the sine- and cosine transform, respectively. In theoretical model calculations the present method is considerably more efficient than the Fast Fourier Transform (FFT).
Type of Medium:
Electronic Resource
URL:
http://dx.doi.org/10.1111/j.1365-2478.1979.tb01005.x
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