Windows-based line-by-line radiative transfer computation using Mathcad

Transcription

1 Windows-based line-by-line radiative transfer computation using Mathcad I. Introduction Nobuyuki Uemura Fujitsu FIP Corporation, Tokyo, Japan MCLBL (tentative name) is a new Windows-based software environment for line-by-line radiative transfer computation. Because it is built on Mathcad (MathSoft's software for mathematical and technical calculations), it receives full benefits of the powerful features of Mathcad including: An intuitive and easy-to-use user interface is realized. Text, mathematical expressions, and graphs can be integrated into a single worksheet. (Actually, this entire poster was prepared with Mathcad.) Mathematical expressions are typed and displayed in standard mathematical notation. Each expression in a worksheet is evaluated from top to bottom, just like a scripting language. The results from calculation can be directly used for creating graphs or plots. In 'automatic mode' changes made by a user go into effect immediately, affecting the rest of the document (including graphs and plots). Many very useful built-in mathematical functions are provided. Mathcad is capable of calling user-defined functions compiled in external DLL files (we often call them plug-ins ). Users can add their special functions if the built-in functions are insufficient. In particular, MCLBL utilizes the last feature to provide users with a line-by-line computation function. The line-by-line plug-in uses a special algorithm to accelerate the speed of line-by-line computations by factors of 5- without significant loss of accuracy. Although we have also created such plug-ins for MT_CKD continuum model by S.A. Clough et al. and CO 2 line mixing correction by J.-M. Hartmann et al., most computations including complex ones can be performed without special plug-ins. (For example, later we will see that Curtis-Godson pressure/temperature can be computed only with the built-in functions.) A simulation of infrared transmission spectra observed in the solar occultation geometry by a Fourier transform spectrometer is presented as an example. II. Rapid line-by-line algorithm The rapid line-by-line computation algorithm is the heart of MCLBL. It is based on a paper by A. Uchiyama (992) with additional improvements to increase accuracy and speed. Among other things, it has the following characteristics:

2 Line-by-line computations are performed directly on the equidistant output grids specified by users. Voigt line shape is decomposed into several sub-functions and sub-functions with large half-widths are evaluated at large intervals (Fig. ). The exact first derivatives of the Voigt function are used for smoothly connecting the cubic function with the wings of the Voigt function. The rapid line-by-line plug-in currently uses a stripped-down version of R.J. Wells' HUMDEV code for computing Voigt function values and their derivatives. Other codes for computing the complex error function (erfc) and the Faddeeva function defined as w( z) i = π exp( t z t 2 ) dt = exp( z 2 )erfc( iz) (Eq. ) can also be used with the following relations: K ( x, y) = Re( w( z)), K( x, y) x = 2[ y Im( w( z)) xre( w( z)) ]. (Eq. 2) Interpolation is performed after summing the contribution from all the absorption lines for each level of sub-function. Hence the computing cost of interpolation is independent of the total number of absorption lines. Cubic spline interpolation with exact first derivatives for any given function is done by very simple arithmetic (Fig.2 & Eq.3) on equidistant grids. sub-lorentzian correction of CO 2 (Cousin et al., 985) is implemented. sub-function with the largest width cubic function Voigt function y y y a b c = y 3 y 8 ( x x ) y 9 ( x x) y (Eq. 3) sub-functions with intermediate widths cubic function Voigt function cubic function y a 8 y b = 24 6( x x) y c y 5 y 4 ( x x) y 3 ( x x ) y y = f(x) sub-function with the smallest width y y a y b yc The line centre doesn t necessarily coincide with grids. Voigt function cubic function y x x x Fig. Decomposed Voigt line shape Fig. 2 Cubic spline interpolation with exact first derivatives

3 III. Examples (A) NO 2 cross section and gas cell transmittance Cross section mol molecule ID for NO2 pres 5 pressure [hpa] temp 25 temperature [K] vsta 55 vend 67 vint.5 wavenumber range and intervals [cm - ] Xsec lbyl mol, vsta, vend +, vint, pres, temp rapid line-by-line computations i.. rows Xsec v i vsta + vint i Cross Section [cm2/molecule] Wavenumber [cm-] Fig. 3 Cross section of NO2

4 Xref lbyl2 mol, 6, 6, vint, pres, temp standard line-by-line computations with a highly accurate algorithm for Voigt by Shippony & Reed as a reference j.. rows Xref v2 j 6 + vint j.5 Relative error [%] Wavenumber [cm-] Fig. 4 Relative error of cross section We are working on a refinement of the algorithm which will further reduce the errors by about a factor of /2. Loschmidt Loschmidt number [molecule/cm 3 ] rhof Loschmidt conversion factor from p/t to number density Column rhof pres 5 temp column for 5 cm cell [molecule/cm 2 ] Tm e Xsec Column transmittance res. vints.25 res spectral resolution [cm - ] Ts scan Tm, vsta, vend +, vint, vsta, vend, vints, res convolution with triangle function i.. rows Ts v i vsta + vints i

5 Wavenumber [cm-] Fig. 5 (NO 2 ) (B) Simulation of infrared transmission spectra observed in the solar occultation geometry by a Fourier transform spectrometer Constants and other setup R e Earth radius [km] ilo function for interval finding indx v, x ihi while return rows v ihi ilo > ilo + ihi i floor 2 ihi i if x < v i ilo i otherwise ilo Input profile data for pressure, temperature, and gas concentration from an external file STD READPRN ("std.prn") US Standard Atmosphere (plain text format) profiles at 5 discrete points between [-2km] H STD P STD T STD 2 ( ) 6 X submatrix STD,, rows STD, 3, cols STD

6 Definitions of continuous functions for pressure, number density, temperature, and gas concentrations i indx( H, h) ph xh, j h H i h H i+ h h h h P i+ return P i P i + x( h, ) return if j = i indx H, h h H i h H i+ jj j X i+, jj return X i, jj X i, jj h h h h rho h th i indx( H, h) h H i h H i+ P i return rhof T i rhof p h rho( h) P i+ T i+ P i T i h h h h Refractive index of the air 8 Np σ, p, t, χh σ p σ χh p σ 2 + ( t 273.5) refractive index of air - by Peck & Reeder (972) (function of wavenumber, pressure, temperature, and water vapor) + Np Nh,.724 ph, th, xh (, ) refractive Index of the air at.724 micro meters (function of altitude) Integrals along with light path (slant column, Curtis-Godson mean values) Nh ( R e + h) nr h Rhox rg, i, j H i + xh (, j) rho( h) nr( h) d + x( h, ) h 5 nr( h) 2 rg 2 H i Pave rg, i, j H i + xh (, j) ph rho( h) nr( h) d + x( h, ) h 5 nr( h) 2 rg 2 H i Rhox( rg, i, j)

7 Tave rg, i, j rhof H i H i + xh (, j) ph nr( h) h + x h, nr( h) 2 rg 2 Rhox rg, i, j d 5 Monochromatic transmittance vsta 52 vend 64 vint.2 wavenumber range and intervals [cm - ] L TH H TH =. tangent height [km] rg nr( TH) constant of Snell's Law in spherical geometry H2O 44 i = L ( ) ( 2 Rhox( rg, i, ) ) lbyl, vsta, vend +, vint, Pave rg, i,, Tave rg, i, optical depth for H 2 O CO2 CH4 44 i = L 44 i = L optical depth for CO 2 ( ) ( 2 Rhox( rg, i, 2) ) lbyl 2, vsta, vend +, vint, Pave rg, i, 2, Tave rg, i, 2 ( ) ( 2 Rhox( rg, i, 6) ) lbyl 6, vsta, vend +, vint, Pave rg, i, 6, Tave rg, i, 6 optical depth for CH 4 OD H2O + CO2 + CH4 optical depth for the gases Tg e OD transmittance for the gases Norton-Beer apodization ln( rows( Tg) ) ceil ln( 2) nfft 2 size of FFT i rows Tg Tg i.. nfft zero filling IGM FFT( Tg) FFT to get interferogram

8 MPD 5 Maximum Path Difference [cm] dx nfft vint MPD ncut floor + dx i.. ncut interferogram cutoff at MPD TMP i IGM i IGM TMP Apodize x x 2 x MPD MPD Norton-Beer medium apodization function x i dx i IGM i IGM i Apodize x i apodization ln( rows( IGM) ) ceil ln( 2) nfft size of inverse FFT i rows IGM IGM i.. nfft2 zero filling Tg IFFT( IGM) inverse FFT to get transmittance back vints dx 2 nfft2 new wavenumber intervals vend vsta i.. floor vints +. v v i vsta + vints i truncation of the unnecessary part TMP i Tg i Tg TMP

9 Wavenumber Fig. 6 of gases near at km (about.7 cm - resolution) Aerosols, Rayleigh scattering, and Continuum 44 ( ) cntnm vsta, vend, vints, Pave rg, i,, Tave rg, i,, 2 Rhox rg, i,, 2Rhox rg, i,, 2Rhox rg, i, 2,, 4 i = L Tc e water vapor continuum, CO2 continuum, and Rayleigh scattering Ta.8 transmittance due to aerosols is set to a constant for simplicity Ttot ( Tg Tc) Ta transmittance in total Wavenumber Fig. 7 at km

10 Wavenumber Fig. 8 near.6 micro meters Wavenumber Fig. 9 (68 cm cm-) Producing variants of a plot (like making Fig.8 and Fig.9 from Fig 7) is merely performing 'Cut&Paste' followed by a little change of graph properties.

11 Extras (Ray tracing) nr( h) 2 dh = nr( h) 2 rg 2 path length [km] between km and km layer boundaries for a light path with TH= km. nr TH θ asin θ nr( ) deg = zenith angle at space = km [degree] deg 44 H i+ rg dh i = L ( R e + h) nr( h) 2 rg 2 H i + θ 9 =.23 bending angle [degree] IV. Conclusion MCLBL is not a stand-alone software but a set of Mathcad plug-in DLLs and sample worksheets (or e-books) that offers users a new approach for radiative transfer computation. Among other things, the plug-in DLLs contain a special line-by-line computation algorithm * that realizes the interactive line-by-line radiative transfer computations. Fairly complex projects can be done with just a small number of specially created external functions because of Mathcad's powerful built-in functions. Actually, the only external plug-in functions used in the examples are 'lbyl' (rapid line-by-line), 'scan', and 'cntnm'. The biggest advantage of MCLBL is probably its flexibility. The sample worksheets can be easily modified and customized to create new ones. V. Outlook * patent pending MCLBL is not yet complete and is currently in the process of development. We plan to release it as a shareware product in near future. References Mathcad ( MathSoft ( S.A. Clough et al., MT_CKD ( R. Rodrigues, et al., JQSRT 6, pp (999) A. Uchiyama, JQSRT 47, pp (992) R.J. Wells, JQSRT, 62, pp (999) ( C. Cousin et al., Appl. Opt., Vol.24, No.22, pp (985) Z. Shippony and W.G. Reed, JQSRT 5, pp (993); JQSRT 78, p.255 (23) E.R. Peck and K. Reeder, J. Opt. Soc. Am., Vol.62, No.8, pp (972) C.D. Rodgers, "Inverse Methods for Atmospheric Soundings", World Scientific Publ.