Approximation to the Lorentzian coefficient for efficient calculation of transmittance profiles

Joseph H. Pierluissi; Glenn A. Gibson; Richard B. Gomez

doi:10.1364/AO.17.001425

Applied Optics
Vol. 17,
Issue 9,
pp. 1425-1431
(1978)
•https://doi.org/10.1364/AO.17.001425

Approximation to the Lorentzian coefficient for efficient calculation of transmittance profiles

Joseph H. Pierluissi, Glenn A. Gibson, and Richard B. Gomez

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Joseph H. Pierluissi, Glenn A. Gibson, and Richard B. Gomez, "Approximation to the Lorentzian coefficient for efficient calculation of transmittance profiles," Appl. Opt. 17, 1425-1431 (1978)

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Abstract

A major problem in the calculation of line-by-line profiles of atmospheric transmittance lies in the excessive computational times associated with the evaluation of the Lorentzian coefficient for every gas, wavenumber, line, temperature, and pressure along the path. An approach to the solution of this problem is presented, whereby use is made of an approximating function that allows for the quantities involving temperature and pressure to be factored out of the wavenumber-dependent terms. Although the approximating function is restricted to wavenumbers farther than about a line halfwidth at STP from the line center, a numerical procedure is presented for dealing with the remaining few cases. This approach results in a significant reduction in the number of arithmetic operations from the use of the exact coefficient and generally yields a transmittance with a numerical accuracy of four significant figures or better. An application is made to five Nimbus 6 center-filter frequencies in the 4.3-μm CO₂ band for H₂O, CO₂, N₂O, and CO in a 33-level atmosphere with average computational times reduced by a factor of over 9.

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MATRIX	DEPENDENCE OF MATRIX ELEMENTS
MATRIX	DIMENSION	ABSORBER	LINE	WAVENUMBER	TEMPERATURE	PRESSURE	LEVEL
D_j	J × 1	X		X	X	X	X
W	1 × N				X		X
E_j	N × I_j	X	X
S_j	I_j × I_j^*	X	X
F_j	I_j × M	X	X	X
G_j	M × 1	X			X	X	X

NO.	FUNCTION	ADDITIONS	SUBSTRACTIONS	MULTIPLICATION	DIVISIONS	EXPONENTIATIONS	FRACTIONAL POWERS	INTERPOLATIONS	MULTIPLYING FACTORS
1	X_j and $X_{j}^{2}$			2J	2		J		L
2	M_j Q_j			2J			J	2J	L
3	$exp {\frac{E_{ij}}{{kT}_{o}} (1 - w)}$		1	I	1	I			L
4	$\frac{X_{j}}{Y_{ij}^{2} + X_{j}^{2}}$	I	I	I	2I				VL
5	$\frac{C}{g} \sum_{j} M_{j} K_{j}$	I−1		2I+J+1					VL
6	E_j		I	(2N−3)I					o
7	S_j	I		3I	I+I	I			o
8	E_jS_j			NI					o
9	F_j	I	I	3I	3I				V
10	(E_jS_j)F_j	NM(I−J)		NMI					V
11	X_j and $X_{j}^{2}$			2J	2		J		L
12	G_j	J	J	MJ	J				L
13	(E_jS_jF_j)G_j	N(M−1)		NM					VL
14	D_j			4J	1	J		2J	L
15	$\sum_{j} d_{j} (E_{j} S_{j} F_{j} G_{j})$	N(J−1)			NJ				VL
16	W			1	N−2				L
17	$\frac{C}{g} W \sum_{j} d_{j} (E_{j} S_{j} F_{j} G_{j})$	N−1			N+1				VL

NIMBUS 6 CHANNEL NUMBER	CENTER (CM⁻¹)	CHANNEL WIDTH (CM⁻¹)	TOTAL LINES	MAXIMUM ABSOLUTE DEVIATION IN LORENTZ COEFFICIENT (ATM. CM)⁻¹				CPU TIME EXACT/APPROX.	ARITHMETIC TIME EXACT/APPROX.
NIMBUS 6 CHANNEL NUMBER	CENTER (CM⁻¹)	CHANNEL WIDTH (CM⁻¹)	TOTAL LINES	H₂O	CO₂	N₂O	CO	CPU TIME EXACT/APPROX.	ARITHMETIC TIME EXACT/APPROX.
11	2190	20.6	993	2.2E–11	1.1E–10	1.3E–5	1.9E–7	9.67	10.10
12	2212	22.5	1058	3.0E–10	9.0E–9	2.6E–3	1.2E–6	9.32	10.23
13	2242	21.6	921	8.9E–12	7.0E–8	3.2E–4	7.1E–9	8.90	9.94
14	2275	35.2	1597	1.3E–9	6.6E–5	5.8E–8	2.0E–13	15.59	10.95
15	2357	23.0	504	3.5E–12	8.9E–5	5.7×10⁻⁹	0.0	5.09	8.43

MATRIX	DEPENDENCE OF MATRIX ELEMENTS
MATRIX	DIMENSION	ABSORBER	LINE	WAVENUMBER	TEMPERATURE	PRESSURE	LEVEL
D_j	J × 1	X		X	X	X	X
W	1 × N				X		X
E_j	N × I_j	X	X
S_j	I_j × I_j^*	X	X
F_j	I_j × M	X	X	X
G_j	M × 1	X			X	X	X

NO.	FUNCTION	ADDITIONS	SUBSTRACTIONS	MULTIPLICATION	DIVISIONS	EXPONENTIATIONS	FRACTIONAL POWERS	INTERPOLATIONS	MULTIPLYING FACTORS
1	X_j and $X_{j}^{2}$			2J	2		J		L
2	M_j Q_j			2J			J	2J	L
3	$exp {\frac{E_{ij}}{{kT}_{o}} (1 - w)}$		1	I	1	I			L
4	$\frac{X_{j}}{Y_{ij}^{2} + X_{j}^{2}}$	I	I	I	2I				VL
5	$\frac{C}{g} \sum_{j} M_{j} K_{j}$	I−1		2I+J+1					VL
6	E_j		I	(2N−3)I					o
7	S_j	I		3I	I+I	I			o
8	E_jS_j			NI					o
9	F_j	I	I	3I	3I				V
10	(E_jS_j)F_j	NM(I−J)		NMI					V
11	X_j and $X_{j}^{2}$			2J	2		J		L
12	G_j	J	J	MJ	J				L
13	(E_jS_jF_j)G_j	N(M−1)		NM					VL
14	D_j			4J	1	J		2J	L
15	$\sum_{j} d_{j} (E_{j} S_{j} F_{j} G_{j})$	N(J−1)			NJ				VL
16	W			1	N−2				L
17	$\frac{C}{g} W \sum_{j} d_{j} (E_{j} S_{j} F_{j} G_{j})$	N−1			N+1				VL

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Applied Optics