Abstract

Assumptions made in the analysis of both Raman lidar measurements of aerosol extinction and differential absorption lidar (DIAL) measurements of an absorbing species are tested. Statistical analysis techniques are used to enhance the estimation of aerosol extinction and aerosol extinction error that is usually handled using a linear model. It is determined that the most probable extinction value can differ from that of the linear assumption by up to 10% and that differences larger than 50% can occur in the calculation of extinction error. Ignoring error in the number density alters the calculated extinction by up to 3% and that of extinction error by up to 10%. The preceding results were obtained using the least-squares technique. The least-squares technique assumes that the data being regressed are normally distributed. However, the quantity that is usually regressed in aerosol extinction and DIAL calculations is not normally distributed. A technique is presented that allows the required numerical derivative to be determined by regressing only normally distributed data. The results from this technique are compared with the usual procedure. The same concerns raised here regarding appropriate choice of a model in the context of aerosol extinction calculations should apply to DIAL calculations of absorbing species such as water vapor or ozone as well because the numerical derivative that is required is identical.

© 1999 Optical Society of America

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References

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  1. A. Ansmann, M. Riebesell, C. Weitkamp, “Measurement of atmospheric aerosol extinction profiles with a Raman lidar,” Opt. Lett. 15, 746–748 (1990).
    [CrossRef] [PubMed]
  2. T. J. McGee, D. Whiteman, R. Ferrare, J. Butler, J. Burris, “STROZ LITE: stratospheric ozone lidar trailer experiment,” Opt. Eng. 30, 31–39 (1991).
    [CrossRef]
  3. R. Measures, Laser Remote Sensing Fundamentals and Applications (Wiley-Interscience, London, 1984).
  4. R. J. Barlow, Statistics: A Guide to the Use of Statistical Methods in the Physical Sciences (Wiley, New York, 1989).
  5. R. V. Hogg, E. A. Tanis, Probability and Statistical Inference, 4th ed. (MacMillan, New York, 1993).
  6. J. R. Taylor, An Introduction to Error Analysis—The Study of Uncertainties in Physical Measurements (University Science, Mill Valley, Calif., 1982).
  7. D. N. Whiteman, W. F. Murphy, N. W. Walsh, K. D. Evans, “Temperature sensitivity of an atmospheric Raman lidar system based on a XeF excimer laser,” Opt. Lett. 18, 247–249 (1993).
    [CrossRef]

1993 (1)

1991 (1)

T. J. McGee, D. Whiteman, R. Ferrare, J. Butler, J. Burris, “STROZ LITE: stratospheric ozone lidar trailer experiment,” Opt. Eng. 30, 31–39 (1991).
[CrossRef]

1990 (1)

Ansmann, A.

Barlow, R. J.

R. J. Barlow, Statistics: A Guide to the Use of Statistical Methods in the Physical Sciences (Wiley, New York, 1989).

Burris, J.

T. J. McGee, D. Whiteman, R. Ferrare, J. Butler, J. Burris, “STROZ LITE: stratospheric ozone lidar trailer experiment,” Opt. Eng. 30, 31–39 (1991).
[CrossRef]

Butler, J.

T. J. McGee, D. Whiteman, R. Ferrare, J. Butler, J. Burris, “STROZ LITE: stratospheric ozone lidar trailer experiment,” Opt. Eng. 30, 31–39 (1991).
[CrossRef]

Evans, K. D.

Ferrare, R.

T. J. McGee, D. Whiteman, R. Ferrare, J. Butler, J. Burris, “STROZ LITE: stratospheric ozone lidar trailer experiment,” Opt. Eng. 30, 31–39 (1991).
[CrossRef]

Hogg, R. V.

R. V. Hogg, E. A. Tanis, Probability and Statistical Inference, 4th ed. (MacMillan, New York, 1993).

McGee, T. J.

T. J. McGee, D. Whiteman, R. Ferrare, J. Butler, J. Burris, “STROZ LITE: stratospheric ozone lidar trailer experiment,” Opt. Eng. 30, 31–39 (1991).
[CrossRef]

Measures, R.

R. Measures, Laser Remote Sensing Fundamentals and Applications (Wiley-Interscience, London, 1984).

Murphy, W. F.

Riebesell, M.

Tanis, E. A.

R. V. Hogg, E. A. Tanis, Probability and Statistical Inference, 4th ed. (MacMillan, New York, 1993).

Taylor, J. R.

J. R. Taylor, An Introduction to Error Analysis—The Study of Uncertainties in Physical Measurements (University Science, Mill Valley, Calif., 1982).

Walsh, N. W.

Weitkamp, C.

Whiteman, D.

T. J. McGee, D. Whiteman, R. Ferrare, J. Butler, J. Burris, “STROZ LITE: stratospheric ozone lidar trailer experiment,” Opt. Eng. 30, 31–39 (1991).
[CrossRef]

Whiteman, D. N.

Opt. Eng. (1)

T. J. McGee, D. Whiteman, R. Ferrare, J. Butler, J. Burris, “STROZ LITE: stratospheric ozone lidar trailer experiment,” Opt. Eng. 30, 31–39 (1991).
[CrossRef]

Opt. Lett. (2)

Other (4)

R. Measures, Laser Remote Sensing Fundamentals and Applications (Wiley-Interscience, London, 1984).

R. J. Barlow, Statistics: A Guide to the Use of Statistical Methods in the Physical Sciences (Wiley, New York, 1989).

R. V. Hogg, E. A. Tanis, Probability and Statistical Inference, 4th ed. (MacMillan, New York, 1993).

J. R. Taylor, An Introduction to Error Analysis—The Study of Uncertainties in Physical Measurements (University Science, Mill Valley, Calif., 1982).

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Figures (14)

Fig. 1
Fig. 1

Aerosol extinction calculated using both a linear and a quadratic model for the logarithm of the ratio in expression (2). The first plot shows the standard linear technique; the second plot shows a second-order polynomial fit of these same data. No error was assigned to the radiosonde number density data. The notation (2, 0%) indicates that expression (2) was regressed and that 0% error was assigned to the number density data.

Fig. 2
Fig. 2

Comparison of extinction values and extinction error of the data in Fig. 1. On the left is the ratio of the [extinction error using the quadratic regression for the derivative of expression (2)] and on the right is [the extinction error using the linear regression to determine the derivative of expression (2)]. The extinction values differ by up to ±10% at 3 km. However, the aerosol optical depth between 1 and 3 km agrees to better than 1% between the two techniques. On the right is shown the ratio of these extinction errors where the values can differ by up to 50% or more. On average, the error in determining the slope using the quadratic model is larger than the error in determining the slope using the linear model, with the quadratic model yielding an error in aerosol optical depth 14% higher than the linear model.

Fig. 3
Fig. 3

χ2 and cumulative probability plots for the regressions of Figs. 1 and 2. In general, a χ2 minimum should approximately equal the degrees of freedom in a regression. Using the linear model on the left, the number of degrees of freedom is 3. Using the quadratic model on the right, the number of degrees of freedom is 2. A cumulative probability of approximately 0.5 indicates that a reasonable χ2 minimum value was achieved.

Fig. 4
Fig. 4

Effect of the error attributed to radiosonde number density on the calculation of extinction and extinction error is shown here. On the left is the ratio of (extinction calculated with a linear model and assigning 1.25% to the number density error) and (the extinction calculated with a linear model and assigning 0% to the number density error). The two calculations differ by ±3% over the range of the profile. In the comparison of the extinction errors on the right, it can be seen that the effect of using errors in the number density are mainly confined to the lowest part of the profile where the two techniques differ by up to 10%. This is due to the fact that the lidar data errors are generally much smaller than the radiosonde error in the lowest part of the profile, but they are larger above 2 km and thus become dominant. Aerosol optical depth and error in aerosol optical depth agree to better than 1% between the two techniques.

Fig. 5
Fig. 5

Effect of using 1.25% number density error on χ2 and cumulative χ2 probability are shown here. Comparing these plots with those of Fig. 3 shows that including error for the number density yields more probable results between 1 and 2 km. Above 2 km where lidar error dominates the number density error, there is little difference between the two.

Fig. 6
Fig. 6

Comparison of aerosol extinction and extinction error using a linear model on expression (3) to the standard technique. The extinction values differ by up to 5% in the upper portion of the profile. Extinction errors differ by up to 40% in the lower part of the profile. Aerosol optical depth calculated with the two techniques agrees to within 1% whereas the linear regression of expression (3) yields an error in aerosol optical depth that is 5% more than the standard technique. This is choice (3) for the composite model. Plots of χ2 and cumulative χ2 probability are shown for reference.

Fig. 7
Fig. 7

Comparison of aerosol extinction and extinction error using a quadratic model on expression (3) to the standard technique. The extinction values differ by up to 10% in the upper portion of the profile whereas the errors differ by up to 50%. Aerosol optical depth calculated with the two techniques agrees to within 1% whereas the quadratic regression yields an error in aerosol optical depth that is 2% less than the standard technique. The quadratic model used on expression (3) is choice (4) for the composite model. Plots of χ2 and cumulative χ2 probability are shown for reference.

Fig. 8
Fig. 8

Composite profile uses the results of one of four different models at each height. The model used is shown in the plot on the left whereas the total resulting cumulative χ2 probability is shown on the right. Comparison with Figs. 5 7 shows the improvement of the cumulative probability over any of the individual models.

Fig. 9
Fig. 9

Comparison of the four-model composite to the standard technique. The comparison of extinction is on the left and the comparison of extinction error is on the right. The extinction values differ by up to 10% at the top of the profile whereas the errors differ by up to 40%. Aerosol optical depth and error in aerosol optical depth agree to within 1% between the two techniques.

Fig. 10
Fig. 10

Same as Fig. 9 except that 1.25% is used for number density error instead of 0%. The extinction values differ by up to 9% and the extinction errors differ by up to 40%. Aerosol optical depth and error in aerosol optical depth agree to within 1%.

Fig. 11
Fig. 11

Gaussian composite profile that uses choices (3) and (4) from the set of four models. The choice of model as a function of height and the cumulative χ2 probability are shown.

Fig. 12
Fig. 12

Comparison of the two-model Gaussian composite profile to the standard technique. The techniques agree to within 5% in the calculation of extinction, but the errors differ by up to 60%. Aerosol optical depth calculated by the two techniques agrees to within 1%. The error in aerosol optical depth is 3% less using the two-model composite.

Fig. 13
Fig. 13

Same as Fig. 12 except that 1.25% is used for the number density error instead of 0%. Differences between the two techniques are less than approximately 3% over the range of the profile. Errors differ by up to 40%. Aerosol optical depth calculated by the two techniques agrees to within 1%. The error in aerosol optical depth is 3% less using the two-model composite.

Fig. 14
Fig. 14

Comparison of the four-model composite with the two-model Gaussian composite. The techniques differ by up to 8% at the upper end of the profile whereas the errors are in good agreement except in the lowest 0.5 km where they differ by up to 50%.

Equations (29)

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saerλL, z=ddzlnNzz2Pz-smolλL, z-smolλR, z1+λLλR.
ddzlnNzz2Pz=
1Nzddz Nz-1z2Pzddzz2Pz.
Pyi; a=1σi2πexp-yi-fxi; a2/2σi2.
χ2=i=1Nyi-fxi; a2σi2=i=1Nyimeasured-yiidealmeasurement_errori2.
fz=gz=b,     σfz2=σb2.
σfx2fx2=σu2u2+σv2v2,
σfx2=a σuu2.
σgz2=σPz2Pz2+σNz2Nz2.
σgz2=σPz2Pz2.
fz=gz=b+2cz,σfz2=σb2+4z2σc2+4zσbc2,
Pxx2; v=2-ν/2Γν/2 xν-2 exp-x22.
Pχχ2; v=χ2 Pxx2; vdx2.
χ2=i=1Nyi-fzi; a2σi2.
χ2=y˜-f˜V-1y-f,
V=varacovabcovaccovbavarbcovbccovcacovcbvarc=σa2σab2σac2σba2σb2σbc2σca2σcb2σc2,
fz0; a=r crzar.
f=Ca, χ2=y˜-ãC˜V-1y-Ca,
aˆ=C˜V-1C-1C˜V-1y, Vaˆ=C˜Vy-1C-1,
mˆ=zy¯-zy¯z2¯-z¯2, Vmˆ=σm2=σ2Nz2¯-z¯2, bˆ=y-mˆz¯, Vbˆ=σb2=σ2z2¯Nz2¯-z¯2=Vmˆz2¯.
Gki=fkxi,
Vf=GVxG˜.
G=fza, fzb, fzc.
Vfz=fza, fzb, fzcσa2σab2σac2σba2σb2σbc2σca2σcb2σc2fzafzbfzc,
Vfz=σfz2=σa2fza2+σb2fzb2+σc2fzc2+2σab2fzafzb+2σac2fzafzc+2σbc2fzbfzc.
fz=gz=b,    σfz2=σb2.
fz=gz=b+2cz, σfz2=σb2+4z2σc2+4zσbc2.
fz=gzgz-hzhz=ba+bz-ec+ez σfz2=1a+bz4σa2b2+σb2a2-2σab2ab+1c+ez4σc2e2+σe2c2-2σce2ce.
fz=gzgz-hzhz=b+2cza+bz+cz2-e+2fzd+ez+fz2 σfz2=1a+bz+cz24σa2b+2cz2+σb2a-cz22+σc22a+bz2z2-2σab2b+2cza-cz2-2σac22a+bzb+2czz+2σbc22a+bz×a-cz2z+1d+ez+fz24σd2e+2fz2+σe2d-fz22+σf22d+ez2z2-2σde2×e+2fzd-fz2-2σdf22d+eze+2fzz+2σef22d+ezd-fz2z.

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