Abstract

An approach to approximating the instrument response for an unapodized interferometer is presented. The approach comprises functions that are local enough in frequency space (no more than five wave numbers) that one can use the Planck function at a single frequency to calculate the radiance at a given frequency and atmospheric pressure level, and it is well behaved (transmittances change monotonically from 1.0 to 0.0), so existing transmittance calculation procedures can be used. It is faster than calculating radiances at a high resolution, doing a Fourier transform, and then doing a second transform, and it produces brightness temperatures that agree with exact values to better than the 0.01 K that is due to errors in the approximation. The approach is accurate enough and fast enough to be used for calculating unapodized radiances from an interferometer. It also can be used to calculate transmittances as well as radiances.

© 1998 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. L. M. McMillin, L. J. Crone, T. J. Kleespies, “Atmospheric transmittance of an absorbing gas. 5. Improvements to the OPTRAN approach,” Appl. Opt. 34, 8396–8399 (1995).
    [CrossRef] [PubMed]
  2. L. M. McMillin, L. J. Crone, M. D. Goldberg, T. J. Kleespies, “Atmospheric transmittance of an absorbing gas. 4. OPTRAN: a computationally fast and accurate transmittance model for absorbing gases with fixed and with variable mixing ratios at variable viewing angles,” Appl. Opt. 34, 6269–6274 (1995).
    [CrossRef] [PubMed]
  3. L. M. McMillin, H. E. Fleming, “Atmospheric transmittance of an absorbing gas: a computationally fast and accurate transmittance model for absorbing gases with constant mixing ratios in inhomogeneous atmospheres,” Appl. Opt. 15, 358–363 (1976).
    [CrossRef] [PubMed]
  4. H. E. Fleming, L. M. McMillin, “Atmospheric transmittance of an absorbing gas. 2. A computationally fast and accurate transmittance model for slant paths at different zenith angles,” Appl. Opt. 16, 1366–1370 (1977).
    [CrossRef] [PubMed]
  5. L. M. McMillin, H. E. Fleming, M. L. Hill, “Atmospheric transmittance of an absorbing gas. 3. A computationally fast and accurate transmittance model for absorbing gases with variable mixing ratios,” Appl. Opt. 18, 1600–1606 (1979).
    [CrossRef] [PubMed]
  6. M. P. Weinreb, A. C. Neuendorffer, “Method to apply homogeneous-path transmittance models to inhomogeneous atmospheres,” J. Atmos. Sci. 30, 662–666 (1973).
    [CrossRef]
  7. J. Susskind, J. Rosenfield, D. Rueter, “An accurate radiative transfer model for use in the direct inversion physical inversion of HIRS2 and MSU temperature sounding data,” J. Geophys. Res. 88, 8550–8568 (1983).
    [CrossRef]
  8. R. J. Bell, Introductory Fourier Transform Spectroscopy (Academic, New York, 1972).
  9. U.S. Committeeon Extension to the Standard Atmosphere, U.S. Standard Atmosphere and Supplements, 1976 (U.S. Government Printing Office, Washington, D.C., 1976).
  10. R. W. Hamming, Digital Filters, 2nd ed. (Prentice-Hall, Englewood Cliffs, N.J., 1977).
  11. The DDM was developed by J. Susskind, 5March1997.

1995 (2)

1983 (1)

J. Susskind, J. Rosenfield, D. Rueter, “An accurate radiative transfer model for use in the direct inversion physical inversion of HIRS2 and MSU temperature sounding data,” J. Geophys. Res. 88, 8550–8568 (1983).
[CrossRef]

1979 (1)

1977 (1)

1976 (1)

1973 (1)

M. P. Weinreb, A. C. Neuendorffer, “Method to apply homogeneous-path transmittance models to inhomogeneous atmospheres,” J. Atmos. Sci. 30, 662–666 (1973).
[CrossRef]

Bell, R. J.

R. J. Bell, Introductory Fourier Transform Spectroscopy (Academic, New York, 1972).

Crone, L. J.

Fleming, H. E.

Goldberg, M. D.

Hamming, R. W.

R. W. Hamming, Digital Filters, 2nd ed. (Prentice-Hall, Englewood Cliffs, N.J., 1977).

Hill, M. L.

Kleespies, T. J.

McMillin, L. M.

Neuendorffer, A. C.

M. P. Weinreb, A. C. Neuendorffer, “Method to apply homogeneous-path transmittance models to inhomogeneous atmospheres,” J. Atmos. Sci. 30, 662–666 (1973).
[CrossRef]

Rosenfield, J.

J. Susskind, J. Rosenfield, D. Rueter, “An accurate radiative transfer model for use in the direct inversion physical inversion of HIRS2 and MSU temperature sounding data,” J. Geophys. Res. 88, 8550–8568 (1983).
[CrossRef]

Rueter, D.

J. Susskind, J. Rosenfield, D. Rueter, “An accurate radiative transfer model for use in the direct inversion physical inversion of HIRS2 and MSU temperature sounding data,” J. Geophys. Res. 88, 8550–8568 (1983).
[CrossRef]

Susskind, J.

J. Susskind, J. Rosenfield, D. Rueter, “An accurate radiative transfer model for use in the direct inversion physical inversion of HIRS2 and MSU temperature sounding data,” J. Geophys. Res. 88, 8550–8568 (1983).
[CrossRef]

Weinreb, M. P.

M. P. Weinreb, A. C. Neuendorffer, “Method to apply homogeneous-path transmittance models to inhomogeneous atmospheres,” J. Atmos. Sci. 30, 662–666 (1973).
[CrossRef]

Appl. Opt. (5)

J. Atmos. Sci. (1)

M. P. Weinreb, A. C. Neuendorffer, “Method to apply homogeneous-path transmittance models to inhomogeneous atmospheres,” J. Atmos. Sci. 30, 662–666 (1973).
[CrossRef]

J. Geophys. Res. (1)

J. Susskind, J. Rosenfield, D. Rueter, “An accurate radiative transfer model for use in the direct inversion physical inversion of HIRS2 and MSU temperature sounding data,” J. Geophys. Res. 88, 8550–8568 (1983).
[CrossRef]

Other (4)

R. J. Bell, Introductory Fourier Transform Spectroscopy (Academic, New York, 1972).

U.S. Committeeon Extension to the Standard Atmosphere, U.S. Standard Atmosphere and Supplements, 1976 (U.S. Government Printing Office, Washington, D.C., 1976).

R. W. Hamming, Digital Filters, 2nd ed. (Prentice-Hall, Englewood Cliffs, N.J., 1977).

The DDM was developed by J. Susskind, 5March1997.

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (8)

Fig. 1
Fig. 1

Transmittance with interferometric thermal sounder convolution: Transmittances as a function of pressure for a channel showing a region of negative transmittance for one 1976 U.S. standard atmosphere, channel 211, frequency of 740.98461 cm-1.

Fig. 2
Fig. 2

Transmittance with interferometric thermal sounder convolution: Transmittance as a function of pressure for a channel showing a region of transmittance that exceeds 1.0 for one 1976 U.S. standard atmosphere, channel 224, frequency of 749.1063 cm-1.

Fig. 3
Fig. 3

Sinc function (top), the five terms used for its approximation (graphs 2–4, top to bottom), and the error (bottom).

Fig. 4
Fig. 4

True transmittance and transmittance error for the five-term approximation: (a) Transmittance spectrum with sinc function convolution at the surface of 1976 U.S. standard atmosphere. (b) Five-piece functions estimation error.

Fig. 5
Fig. 5

Radiances for the five-term approximation scaled by 2 and 40 at the shorter wavelengths, the error for the reference atmosphere, and the error if no normalization to a reference is performed: (a) Radiances for sinc function convolution for one 1976 U.S. standard atmosphere. (b) Five-term method: from radiances at a 0.01 cm-1 resolution. (c) Five-term method: from radiances at instrument resolution.

Fig. 6
Fig. 6

Brightness temperature, brightness-temperature error for the normalized three-term estimation, and brightness-temperature error for the three-term approximation with the transmittances limited to between 0 and 1: (a) Brightness temperature for the sinc function convolution for one 1976 U.S. standard atmosphere. (b) Three-term estimation error. (c) Three-term estimation error for 0.0 ≤ τ ≤ 1.0.

Fig. 7
Fig. 7

Radiances for the Hamming apodization applied to the sinc function and the difference from the Hamming apodization applied to the true radiance: (a) Radiances for the Hamming function convolution for one 1976 U.S. standard atmosphere. (b) From radiances calculated at the instrument resolution.

Fig. 8
Fig. 8

True radiances for the DDM, noise-equivalent brightness-temperature errors at 250 K with radiances normalized to the reference atmosphere, and noise-equivalent brightness-temperature errors for the nonnormalized case: (a) Radiances for the sinc function convolution for one 1976 U.S. standard atmosphere. (b) Matrix method: from radiances at a 0.01 cm-1 resolution. (c) Matrix method: from radiances at the instrument resolution.

Tables (1)

Tables Icon

Table 1 Instrument Characteristics of an Interferometric Thermal Sounder

Equations (15)

Equations on this page are rendered with MathJax. Learn more.

I ν = B s ν τ s ν + τ s ν 1   B ν d τ ν ,
FWHM = 0.603355 L 1.2 2 L .
I i =   F ν ϕ ν - ν i B s ν τ s ν + τ s ν 1   B ν d τ ν ,
I i B s i τ s i + τ s ν 1   B i d τ i ,
I i = F ν i ¯ ν   ϕ ν - ν i d ν 0 1 B ν i ¯ d τ ν i ¯ ,
τ ν i ¯ = ν l ν u   F ν ϕ j ν - ν i τ ν d ν ν l ν u   F ν ϕ j ν - ν i d ν ,
F ν i ¯ = ν l ν u   F ν ϕ j ν - ν i ν l ν u   ϕ j ν - ν i ,
B ν i ¯ = ν l ν u   F ν ϕ j ν - ν i B ν d ν ν l ν u   F ν ϕ j ν - ν i d ν .
c 1 k = d k / d 0
c 2 k = h k - c 1 k h 0 .
inc x = f 1 x + f 2 x + j = x + 6 j = x + n f 5 x c 1 j - x + f 3 x c 2 j - x + j = x - 6 j = x - m f 4 x c 1 j - x + f 3 x c 2 j - x ,
τ x = τ 1 x + τ 2 x + j = x + 6 j = x + n τ 5 x c 1 j - x + τ 3 x c 2 j - x + j = x - 6 j = x - m τ 4 x c 1 j - x + τ 3 x c 2 j - x ,
r x = r 1 x + r 2 x + j = x + 6 j = x + n r 5 x c 1 j - x + r 3 x c 2 j - x + j = x - 6 j = x - m r 4 x c 1 j - x + r 3 x c 2 j - x .
R H = M × R U ,
R U = M - 1 × R H .

Metrics