Abstract

The processes of discretization, interpolation, and resampling are frequently used in data analysis. Here the formalism of functional spaces is used as a framework for the description and characterization of both the measurement operation and these subsequent processes. The tools provided by this formalism are applied to the problem of resampling of atmospheric volume mixing ratio vertical profiles obtained with limb-sounding measurements. In particular, a resampling method that uses the conservation of the vertical column as a constraint is presented and compared with other methods. The effects of the resampling process in terms of error propagation and loss of vertical resolution are also evaluated.

© 2001 Optical Society of America

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  1. E. H. W. Meijering, K. J. Zuiderveld, M. A. Viergever, “Image reconstruction by convolution with symmetrical piecewise nth-order polynomial kernels,” IEEE Trans. Image Process. 8, 192–201 (1999).
    [CrossRef]
  2. V. Rasche, R. Proksa, R. Sinkus, P. Börnert, H. Eggers, “Resampling of data between arbitrary grids using convolution interpolation,” IEEE Trans. Med. Imaging 18, 385–392 (1999).
    [CrossRef] [PubMed]
  3. J. F. Moreno, J. Melia, “An optimum interpolation method applied to the resampling of NOAA AVHRR Data,” IEEE Trans. Geosci. Remote Sens. 32, 131–150 (1994).
    [CrossRef]
  4. A. C. Lorenc, “Analysis methods for numerical weather prediction,” Q. J. R. Meteorol. Soc. 112, 1177–1194 (1986).
    [CrossRef]
  5. B. V. Khattatov, J. C. Gille, L. V. Lyjak, G. P. Brasseur, V. L. Dvortsov, A. E. Roche, J. W. Waters, “Assimilation of photochemically active species and a case analysis of UARS data,” J. Geophys. Res. 104, 18,715–18,737 (1999).
    [CrossRef]
  6. M. A. Oliver, R. Webster, “Kriging: a method of interpolation for geographical information system,” Int. J. Geog. Inform. Syst. 4, 313–332 (1990).
    [CrossRef]
  7. T. M. Burgess, R. Webster, “Optimal interpolation and isarithmic mapping of soil properties. I. The semivariogram and punctual kriging,” J. Soil Sci. 31, 315–331 (1980).
    [CrossRef]
  8. T. M. Burgess, R. Webster, “Optimal interpolation and isarithmic mapping of soil properties. II. Block kriging,” J. Soil Sci. 31, 333–341 (1980).
    [CrossRef]
  9. European Space Agency, “ENVISAT–MIPAS—an instrument for atmospheric chemistry and climate research,” (European Space Agency, European Space Research and Technology Centre, Noordwijk, The Netherlands, 2000).
  10. S. Twomey, Introduction to the Mathematics of Inversion in Remote Sensing and Indirect Measurements (Elsevier, New York, 1977).
  11. R. E. Kalman, “Algebraic aspects of the generalized inverse of a rectangular matrix,” in Proceedings of Advanced Seminar on Generalized Inverse and Applications, M. Z. Nashed, ed. (Academic, San Diego, Calif., 1976), pp. 111–124.
  12. B. Carli, “A discussion on data reduction in spectrometric measurements,” Infrared Phys. 12, 251–261 (1972).
    [CrossRef]
  13. M. Harwit, N. J. A. Sloane, Hadamard Transform Optics (Academic, New York, 1979).
  14. M. Ridolfi, B. Carli, M. Carlotti, T. v. Clarmann, B. M. Dinelli, A. Dudhia, J.-M. Flaud, M. Höpfner, P. E. Morris, P. Raspollini, G. Stiller, R. J. Wells, “Optimized forward model and retrieval scheme for MIPAS near-real-time data processing,” Appl. Opt. 39, 1323–1340 (2000).
    [CrossRef]
  15. M. Carlotti, B. Carli, “Approach to the design and data analysis of a limb-scanning experiment,” Appl. Opt. 33, 3237–3249 (1994).
    [CrossRef] [PubMed]
  16. B. Carli, M. Ridolfi, P. Raspollini, B. M. Dinelli, A. Dudhia, G. Echle, “Study of the retrieval of atmospheric trace gas profiles from infrared spectra,” (European Space Agency, Munich, Germany, 1998).

2000

1999

E. H. W. Meijering, K. J. Zuiderveld, M. A. Viergever, “Image reconstruction by convolution with symmetrical piecewise nth-order polynomial kernels,” IEEE Trans. Image Process. 8, 192–201 (1999).
[CrossRef]

V. Rasche, R. Proksa, R. Sinkus, P. Börnert, H. Eggers, “Resampling of data between arbitrary grids using convolution interpolation,” IEEE Trans. Med. Imaging 18, 385–392 (1999).
[CrossRef] [PubMed]

B. V. Khattatov, J. C. Gille, L. V. Lyjak, G. P. Brasseur, V. L. Dvortsov, A. E. Roche, J. W. Waters, “Assimilation of photochemically active species and a case analysis of UARS data,” J. Geophys. Res. 104, 18,715–18,737 (1999).
[CrossRef]

1994

J. F. Moreno, J. Melia, “An optimum interpolation method applied to the resampling of NOAA AVHRR Data,” IEEE Trans. Geosci. Remote Sens. 32, 131–150 (1994).
[CrossRef]

M. Carlotti, B. Carli, “Approach to the design and data analysis of a limb-scanning experiment,” Appl. Opt. 33, 3237–3249 (1994).
[CrossRef] [PubMed]

1990

M. A. Oliver, R. Webster, “Kriging: a method of interpolation for geographical information system,” Int. J. Geog. Inform. Syst. 4, 313–332 (1990).
[CrossRef]

1986

A. C. Lorenc, “Analysis methods for numerical weather prediction,” Q. J. R. Meteorol. Soc. 112, 1177–1194 (1986).
[CrossRef]

1980

T. M. Burgess, R. Webster, “Optimal interpolation and isarithmic mapping of soil properties. I. The semivariogram and punctual kriging,” J. Soil Sci. 31, 315–331 (1980).
[CrossRef]

T. M. Burgess, R. Webster, “Optimal interpolation and isarithmic mapping of soil properties. II. Block kriging,” J. Soil Sci. 31, 333–341 (1980).
[CrossRef]

1972

B. Carli, “A discussion on data reduction in spectrometric measurements,” Infrared Phys. 12, 251–261 (1972).
[CrossRef]

Börnert, P.

V. Rasche, R. Proksa, R. Sinkus, P. Börnert, H. Eggers, “Resampling of data between arbitrary grids using convolution interpolation,” IEEE Trans. Med. Imaging 18, 385–392 (1999).
[CrossRef] [PubMed]

Brasseur, G. P.

B. V. Khattatov, J. C. Gille, L. V. Lyjak, G. P. Brasseur, V. L. Dvortsov, A. E. Roche, J. W. Waters, “Assimilation of photochemically active species and a case analysis of UARS data,” J. Geophys. Res. 104, 18,715–18,737 (1999).
[CrossRef]

Burgess, T. M.

T. M. Burgess, R. Webster, “Optimal interpolation and isarithmic mapping of soil properties. I. The semivariogram and punctual kriging,” J. Soil Sci. 31, 315–331 (1980).
[CrossRef]

T. M. Burgess, R. Webster, “Optimal interpolation and isarithmic mapping of soil properties. II. Block kriging,” J. Soil Sci. 31, 333–341 (1980).
[CrossRef]

Carli, B.

M. Ridolfi, B. Carli, M. Carlotti, T. v. Clarmann, B. M. Dinelli, A. Dudhia, J.-M. Flaud, M. Höpfner, P. E. Morris, P. Raspollini, G. Stiller, R. J. Wells, “Optimized forward model and retrieval scheme for MIPAS near-real-time data processing,” Appl. Opt. 39, 1323–1340 (2000).
[CrossRef]

M. Carlotti, B. Carli, “Approach to the design and data analysis of a limb-scanning experiment,” Appl. Opt. 33, 3237–3249 (1994).
[CrossRef] [PubMed]

B. Carli, “A discussion on data reduction in spectrometric measurements,” Infrared Phys. 12, 251–261 (1972).
[CrossRef]

B. Carli, M. Ridolfi, P. Raspollini, B. M. Dinelli, A. Dudhia, G. Echle, “Study of the retrieval of atmospheric trace gas profiles from infrared spectra,” (European Space Agency, Munich, Germany, 1998).

Carlotti, M.

Clarmann, T. v.

Dinelli, B. M.

M. Ridolfi, B. Carli, M. Carlotti, T. v. Clarmann, B. M. Dinelli, A. Dudhia, J.-M. Flaud, M. Höpfner, P. E. Morris, P. Raspollini, G. Stiller, R. J. Wells, “Optimized forward model and retrieval scheme for MIPAS near-real-time data processing,” Appl. Opt. 39, 1323–1340 (2000).
[CrossRef]

B. Carli, M. Ridolfi, P. Raspollini, B. M. Dinelli, A. Dudhia, G. Echle, “Study of the retrieval of atmospheric trace gas profiles from infrared spectra,” (European Space Agency, Munich, Germany, 1998).

Dudhia, A.

M. Ridolfi, B. Carli, M. Carlotti, T. v. Clarmann, B. M. Dinelli, A. Dudhia, J.-M. Flaud, M. Höpfner, P. E. Morris, P. Raspollini, G. Stiller, R. J. Wells, “Optimized forward model and retrieval scheme for MIPAS near-real-time data processing,” Appl. Opt. 39, 1323–1340 (2000).
[CrossRef]

B. Carli, M. Ridolfi, P. Raspollini, B. M. Dinelli, A. Dudhia, G. Echle, “Study of the retrieval of atmospheric trace gas profiles from infrared spectra,” (European Space Agency, Munich, Germany, 1998).

Dvortsov, V. L.

B. V. Khattatov, J. C. Gille, L. V. Lyjak, G. P. Brasseur, V. L. Dvortsov, A. E. Roche, J. W. Waters, “Assimilation of photochemically active species and a case analysis of UARS data,” J. Geophys. Res. 104, 18,715–18,737 (1999).
[CrossRef]

Echle, G.

B. Carli, M. Ridolfi, P. Raspollini, B. M. Dinelli, A. Dudhia, G. Echle, “Study of the retrieval of atmospheric trace gas profiles from infrared spectra,” (European Space Agency, Munich, Germany, 1998).

Eggers, H.

V. Rasche, R. Proksa, R. Sinkus, P. Börnert, H. Eggers, “Resampling of data between arbitrary grids using convolution interpolation,” IEEE Trans. Med. Imaging 18, 385–392 (1999).
[CrossRef] [PubMed]

Flaud, J.-M.

Gille, J. C.

B. V. Khattatov, J. C. Gille, L. V. Lyjak, G. P. Brasseur, V. L. Dvortsov, A. E. Roche, J. W. Waters, “Assimilation of photochemically active species and a case analysis of UARS data,” J. Geophys. Res. 104, 18,715–18,737 (1999).
[CrossRef]

Harwit, M.

M. Harwit, N. J. A. Sloane, Hadamard Transform Optics (Academic, New York, 1979).

Höpfner, M.

Kalman, R. E.

R. E. Kalman, “Algebraic aspects of the generalized inverse of a rectangular matrix,” in Proceedings of Advanced Seminar on Generalized Inverse and Applications, M. Z. Nashed, ed. (Academic, San Diego, Calif., 1976), pp. 111–124.

Khattatov, B. V.

B. V. Khattatov, J. C. Gille, L. V. Lyjak, G. P. Brasseur, V. L. Dvortsov, A. E. Roche, J. W. Waters, “Assimilation of photochemically active species and a case analysis of UARS data,” J. Geophys. Res. 104, 18,715–18,737 (1999).
[CrossRef]

Lorenc, A. C.

A. C. Lorenc, “Analysis methods for numerical weather prediction,” Q. J. R. Meteorol. Soc. 112, 1177–1194 (1986).
[CrossRef]

Lyjak, L. V.

B. V. Khattatov, J. C. Gille, L. V. Lyjak, G. P. Brasseur, V. L. Dvortsov, A. E. Roche, J. W. Waters, “Assimilation of photochemically active species and a case analysis of UARS data,” J. Geophys. Res. 104, 18,715–18,737 (1999).
[CrossRef]

Meijering, E. H. W.

E. H. W. Meijering, K. J. Zuiderveld, M. A. Viergever, “Image reconstruction by convolution with symmetrical piecewise nth-order polynomial kernels,” IEEE Trans. Image Process. 8, 192–201 (1999).
[CrossRef]

Melia, J.

J. F. Moreno, J. Melia, “An optimum interpolation method applied to the resampling of NOAA AVHRR Data,” IEEE Trans. Geosci. Remote Sens. 32, 131–150 (1994).
[CrossRef]

Moreno, J. F.

J. F. Moreno, J. Melia, “An optimum interpolation method applied to the resampling of NOAA AVHRR Data,” IEEE Trans. Geosci. Remote Sens. 32, 131–150 (1994).
[CrossRef]

Morris, P. E.

Oliver, M. A.

M. A. Oliver, R. Webster, “Kriging: a method of interpolation for geographical information system,” Int. J. Geog. Inform. Syst. 4, 313–332 (1990).
[CrossRef]

Proksa, R.

V. Rasche, R. Proksa, R. Sinkus, P. Börnert, H. Eggers, “Resampling of data between arbitrary grids using convolution interpolation,” IEEE Trans. Med. Imaging 18, 385–392 (1999).
[CrossRef] [PubMed]

Rasche, V.

V. Rasche, R. Proksa, R. Sinkus, P. Börnert, H. Eggers, “Resampling of data between arbitrary grids using convolution interpolation,” IEEE Trans. Med. Imaging 18, 385–392 (1999).
[CrossRef] [PubMed]

Raspollini, P.

M. Ridolfi, B. Carli, M. Carlotti, T. v. Clarmann, B. M. Dinelli, A. Dudhia, J.-M. Flaud, M. Höpfner, P. E. Morris, P. Raspollini, G. Stiller, R. J. Wells, “Optimized forward model and retrieval scheme for MIPAS near-real-time data processing,” Appl. Opt. 39, 1323–1340 (2000).
[CrossRef]

B. Carli, M. Ridolfi, P. Raspollini, B. M. Dinelli, A. Dudhia, G. Echle, “Study of the retrieval of atmospheric trace gas profiles from infrared spectra,” (European Space Agency, Munich, Germany, 1998).

Ridolfi, M.

M. Ridolfi, B. Carli, M. Carlotti, T. v. Clarmann, B. M. Dinelli, A. Dudhia, J.-M. Flaud, M. Höpfner, P. E. Morris, P. Raspollini, G. Stiller, R. J. Wells, “Optimized forward model and retrieval scheme for MIPAS near-real-time data processing,” Appl. Opt. 39, 1323–1340 (2000).
[CrossRef]

B. Carli, M. Ridolfi, P. Raspollini, B. M. Dinelli, A. Dudhia, G. Echle, “Study of the retrieval of atmospheric trace gas profiles from infrared spectra,” (European Space Agency, Munich, Germany, 1998).

Roche, A. E.

B. V. Khattatov, J. C. Gille, L. V. Lyjak, G. P. Brasseur, V. L. Dvortsov, A. E. Roche, J. W. Waters, “Assimilation of photochemically active species and a case analysis of UARS data,” J. Geophys. Res. 104, 18,715–18,737 (1999).
[CrossRef]

Sinkus, R.

V. Rasche, R. Proksa, R. Sinkus, P. Börnert, H. Eggers, “Resampling of data between arbitrary grids using convolution interpolation,” IEEE Trans. Med. Imaging 18, 385–392 (1999).
[CrossRef] [PubMed]

Sloane, N. J. A.

M. Harwit, N. J. A. Sloane, Hadamard Transform Optics (Academic, New York, 1979).

Stiller, G.

Twomey, S.

S. Twomey, Introduction to the Mathematics of Inversion in Remote Sensing and Indirect Measurements (Elsevier, New York, 1977).

Viergever, M. A.

E. H. W. Meijering, K. J. Zuiderveld, M. A. Viergever, “Image reconstruction by convolution with symmetrical piecewise nth-order polynomial kernels,” IEEE Trans. Image Process. 8, 192–201 (1999).
[CrossRef]

Waters, J. W.

B. V. Khattatov, J. C. Gille, L. V. Lyjak, G. P. Brasseur, V. L. Dvortsov, A. E. Roche, J. W. Waters, “Assimilation of photochemically active species and a case analysis of UARS data,” J. Geophys. Res. 104, 18,715–18,737 (1999).
[CrossRef]

Webster, R.

M. A. Oliver, R. Webster, “Kriging: a method of interpolation for geographical information system,” Int. J. Geog. Inform. Syst. 4, 313–332 (1990).
[CrossRef]

T. M. Burgess, R. Webster, “Optimal interpolation and isarithmic mapping of soil properties. I. The semivariogram and punctual kriging,” J. Soil Sci. 31, 315–331 (1980).
[CrossRef]

T. M. Burgess, R. Webster, “Optimal interpolation and isarithmic mapping of soil properties. II. Block kriging,” J. Soil Sci. 31, 333–341 (1980).
[CrossRef]

Wells, R. J.

Zuiderveld, K. J.

E. H. W. Meijering, K. J. Zuiderveld, M. A. Viergever, “Image reconstruction by convolution with symmetrical piecewise nth-order polynomial kernels,” IEEE Trans. Image Process. 8, 192–201 (1999).
[CrossRef]

Appl. Opt.

IEEE Trans. Geosci. Remote Sens.

J. F. Moreno, J. Melia, “An optimum interpolation method applied to the resampling of NOAA AVHRR Data,” IEEE Trans. Geosci. Remote Sens. 32, 131–150 (1994).
[CrossRef]

IEEE Trans. Image Process.

E. H. W. Meijering, K. J. Zuiderveld, M. A. Viergever, “Image reconstruction by convolution with symmetrical piecewise nth-order polynomial kernels,” IEEE Trans. Image Process. 8, 192–201 (1999).
[CrossRef]

IEEE Trans. Med. Imaging

V. Rasche, R. Proksa, R. Sinkus, P. Börnert, H. Eggers, “Resampling of data between arbitrary grids using convolution interpolation,” IEEE Trans. Med. Imaging 18, 385–392 (1999).
[CrossRef] [PubMed]

Infrared Phys.

B. Carli, “A discussion on data reduction in spectrometric measurements,” Infrared Phys. 12, 251–261 (1972).
[CrossRef]

Int. J. Geog. Inform. Syst.

M. A. Oliver, R. Webster, “Kriging: a method of interpolation for geographical information system,” Int. J. Geog. Inform. Syst. 4, 313–332 (1990).
[CrossRef]

J. Geophys. Res.

B. V. Khattatov, J. C. Gille, L. V. Lyjak, G. P. Brasseur, V. L. Dvortsov, A. E. Roche, J. W. Waters, “Assimilation of photochemically active species and a case analysis of UARS data,” J. Geophys. Res. 104, 18,715–18,737 (1999).
[CrossRef]

J. Soil Sci.

T. M. Burgess, R. Webster, “Optimal interpolation and isarithmic mapping of soil properties. I. The semivariogram and punctual kriging,” J. Soil Sci. 31, 315–331 (1980).
[CrossRef]

T. M. Burgess, R. Webster, “Optimal interpolation and isarithmic mapping of soil properties. II. Block kriging,” J. Soil Sci. 31, 333–341 (1980).
[CrossRef]

Q. J. R. Meteorol. Soc.

A. C. Lorenc, “Analysis methods for numerical weather prediction,” Q. J. R. Meteorol. Soc. 112, 1177–1194 (1986).
[CrossRef]

Other

European Space Agency, “ENVISAT–MIPAS—an instrument for atmospheric chemistry and climate research,” (European Space Agency, European Space Research and Technology Centre, Noordwijk, The Netherlands, 2000).

S. Twomey, Introduction to the Mathematics of Inversion in Remote Sensing and Indirect Measurements (Elsevier, New York, 1977).

R. E. Kalman, “Algebraic aspects of the generalized inverse of a rectangular matrix,” in Proceedings of Advanced Seminar on Generalized Inverse and Applications, M. Z. Nashed, ed. (Academic, San Diego, Calif., 1976), pp. 111–124.

M. Harwit, N. J. A. Sloane, Hadamard Transform Optics (Academic, New York, 1979).

B. Carli, M. Ridolfi, P. Raspollini, B. M. Dinelli, A. Dudhia, G. Echle, “Study of the retrieval of atmospheric trace gas profiles from infrared spectra,” (European Space Agency, Munich, Germany, 1998).

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Figures (7)

Fig. 1
Fig. 1

Geometrical visualization of the change of representation from a space {g} to a space {h}. The component of vector F′ of space {g} into vector F″ of space {h} can be made by requiring either that F′ and F″ have equal components in space {h} (Fa, classical transformation) or that F′ and F″ have equal components in space {g} (Fb, exuberant transformation).

Fig. 2
Fig. 2

Representation of the functions of the base of the retrieval space (solid lines) and the functions of the base of the user-defined space (dotted lines). The peaks of the triangle functions are located in correspondence with the retrieval altitude grid and the user-defined altitude grid, respectively.

Fig. 3
Fig. 3

HNO3 VMR profiles attained with three resampling techniques (indicated by squares, circles, and triangles), starting from the retrieved profile (filled diamonds). ppb, parts in 109.

Fig. 4
Fig. 4

Effect of three consecutive resampling steps on the original profile. The resampling techniques compared in this figure are the classical transformation and the heuristic one.

Fig. 5
Fig. 5

Comparison of resampling processes performed with the classical transformation and with a transformation that uses the constraint of column conservation after three resampling steps.

Fig. 6
Fig. 6

Vertical resolution as a function of altitude for the classical transformation and for the transformation that uses the constraint of column conservation. The filled circles for the classical resampling and the open circles for the resampling that maintains the columns invariant mark the resampled grid points.

Fig. 7
Fig. 7

Estimate of the change in the measurement error of the resampled discrete profile obtained with both the classical transformation and the transformation that uses the constraint of column conservation.

Equations (24)

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

mi=gi, f,
f=Γc,
m=GTΓc.
c=GTΓ#m,
f=ΓGTΓ#m.
f=Hd.
Condition ahj, f=hj, f,
Condition bgi, f=gi, f,
fa=HHTH#HTGGTG#m,
fb=HGTH#m.
fa=fb=f.
Col = constz1zlimit XgaszPzTzdz,
c=Dxr.
c=Exu.
xu=E-1c=E-1Dxr.
fu=Lxu=LE#DH#Hxr=LE#DH#fr.
fr=HGTH#GTf=Tf.
fu=Pfr=PTf.
P=LLTL#LT=LL#.
P=LE#DH#.
Vu=KVrKT,
C=KKT.
K=LTL#LTH=L#H,
K=E#D.

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