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Rapid and Accurate Calculation of the Voigt Function

Rapid and Accurate Calculation of the Voigt Function. D. Chris Benner Kendra L. Letchworth 9th International HITRAN Conference June 26-28, 2006. Why is this routine needed?. Spectra with higher signal to noise ratios require more accurate analysis routines.

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Rapid and Accurate Calculation of the Voigt Function

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  1. Rapid and Accurate Calculation of the Voigt Function D. Chris Benner Kendra L. Letchworth 9th International HITRAN Conference June 26-28, 2006

  2. Why is this routine needed? • Spectra with higher signal to noise ratios require more accurate analysis routines. • Most fitting programs and simulations perform millions of calculations, so routines must also be fast. • Our routine calculates the Voigt profile to a relative accuracy of 10-6 . • 100 times more accurate than routines such as Drayson & Humlíček but requires 2 to 4 times less calculation time. • Our routine includes the imaginary part of the function for applications to line-mixing. • Option of returning the derivatives of the real Voigt profile with respect to x and y. They are calculated with relative accuracy of less than 10-3, without significant additional calculation time required. • For more information about relative speed and accuracy of various Voigt routines (not including ours) see: F. Schreier, J.Q.S.R.T., Vol. 48, 1992.

  3. The Voigt Profile • v - v0 : distance from line center • αL : Lorentz half-width • αD : Doppler half-width B.H. Armstrong, J.Q.S.R.T., Vol. 7, 1966.

  4. Mathematical Approximations: Gauss-Hermite Quadrature Quadrature Points - vi Quadrature Weights - wi Real Part: Imaginary Part: Expression simplified using: • Symmetry: reduces the number of points calculated from n to (n+1)/2 by taking wi (f(vi) + f(-vi)) and simplifying expression. • Odd-order quadrature: a quadrature point always falls on v=0, giving the simple term at beginning of the equations. • Repeating expressions within equations: (y2+x2+vi2) and (y2+x2+vi2)2-4x2vi2 can be calculated only once.

  5. Mathematical Approximations: Taylor Series Expansion • 3rd order Taylor expansion using a table of pre-computed values of the complex Voigt function (K+iL) and its derivatives. Real Part: • ∂2K/∂x2=-∂2K/∂y2 decreases the number of stored derivatives of the real part (K) from 9 to 6. • Derivatives of the imaginary part (L) can be represented as functions of the derivatives of the real part since ∂L/∂x=-∂K/∂y and ∂L/∂y=∂K/∂x. Imaginary Part: • Δx=x-x0, Δy=y-y0 where (x0,y0) is the closest gridpoint. • A total of 8 values must be stored for each grid point (x0,y0) : K, L, ∂K/∂x, ∂K/∂y, ∂2K/∂x∂y, ∂2K/∂x2, ∂3K/∂x3, ∂3K/∂y3.

  6. Mathematical Approximations:Lagrange Interpolating Polynomials • Used only in small areas when other methods fail, so does not contribute significantly to calculation time. • Employ equal grid point spacing dx and dy. • v1, v2, v3 are the three grid points and ∆v=(v- v2)/dv • Four polynomial interpolations of P(v) below are required for a spline interpolation of real or imaginary part. • A total of eight evaluations is required for one complete function evaluation.

  7. * Note that the Lagrange interpolation regions along the x- and y- axes are so small that they barely appear on the graph.

  8. Programming Techniques • Calculates the Voigt profile for an entire spectral line at one time, removing unnecessary subroutine calls. • Each parameter involving y is calculated only once per spectral line, saving calculation time. • To do this we require equal spacing in wavenumber; a version of the routine called for individual points is available, but not as time efficient. • Interpolation tables stored as binary files on the hard drive and read in only when needed. • All files take up a total of 1.5 MB of memory, a small price to pay for the increase in accuracy and speed.

  9. Accuracy Comparisons Less than 10-4 ↓ ↑ Less than 10-6 J. Humlíček, J.Q.S.R.T., Vol. 27, 1982.

  10. Routines calculating only the Real Part S.R. Drayson, J.Q.S.R.T., Vol.16, 1976. J. H. Pierluissi, J.Q.S.R.T., Vol. 18, 1977. Twitty, Rarig, & Thompson, J.Q.S.R.T.,Vol. 24, 1980. Maximum Error 7x10-4 Maximum Error 1.5x10-2 How bad do some routines get at small y? R. J. Wells, J.Q.S.R.T., Vol. 62, 1999

  11. Speed Comparisons * Humlíček * Drayson calculates only the real part. ** These times are close to the final values, but the routine with derivatives is still undergoing final testing.

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