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Applied Optics

Applied Optics

APPLICATIONS-CENTERED RESEARCH IN OPTICS

  • Editor: Joseph N. Mait
  • Vol. 51, Iss. 33 — Nov. 20, 2012
  • pp: 7945–7952
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New technique for retrieval of atmospheric temperature profiles from Rayleigh-scatter lidar measurements using nonlinear inversion

Jaya Khanna, Justin Bandoro, R. J. Sica, and C. Thomas McElroy  »View Author Affiliations


Applied Optics, Vol. 51, Issue 33, pp. 7945-7952 (2012)
http://dx.doi.org/10.1364/AO.51.007945


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Abstract

The conventional method of calculating atmospheric temperature profiles using Rayleigh-scattering lidar measurements has limitations that necessitate abandoning temperatures retrieved at the greatest heights, due to the assumption of a pressure value required to initialize the integration at the highest altitude. An inversion approach is used to develop an alternative way of retrieving nightly atmospheric temperature profiles from the lidar measurements. Measurements obtained by the Purple Crow lidar facility located near The University of Western Ontario are used to develop and test this new technique. Our results show temperatures can be reliably retrieved at all heights where measurements with adequate signal-to-noise ratio exist. A Monte Carlo technique was developed to provide accurate estimates of both the systematic and random uncertainties for the retrieved nightly average temperature profile. An advantage of this new method is the ability to seed the temperature integration from the lowest rather than the greatest height, where the variability of the pressure is smaller than in the mesosphere or lower thermosphere and may in practice be routinely measured by a radiosonde, rather than requiring a rocket or satellite-borne measurement. Thus, this new technique extends the altitude range of existing Rayleigh-scatter lidars 10–15 km, producing the equivalent of four times the power-aperture product.

© 2012 Optical Society of America

1. Introduction

2. Background

The CH method of atmospheric temperature retrieval from Rayleigh-scatter lidar measurements is based on two assumptions, the first that the atmosphere behaves as an ideal gas and the second that the atmosphere can be considered static and in hydrostatic equilibrium at the spatial-temporal resolution at which the measurements are obtained.

It can be shown using these assumptions that the temperature, Ti at height zi is
Ti=P0MRρi+MRziz0ρ(z)g(z)ρidz,
(1)
where T is determined at the ith discrete measurement point, ρi is the density, g is the acceleration due to gravity, P0 is the seed pressure at the greatest height, R the gas constant for dry air (Jkg1K1), and M the mean mass of air (kg). In the CH method, the atmosphere is assumed to be isothermal to simplify the integration.

The second term in Eq. (1) is independent of the seed pressure value. Thus, the systematic uncertainty from the estimation of P0 affects only the first term of the right-hand side. The seed pressure value can be represented as a sum of the true signal Ptrue and the error ΔP as P0=Ptrue+ΔP. Then Eq. (1) becomes
Ti=Ttrue+ΔT,
(2)
where
Ttrue=MPtrueRρi+MRziz0ρ(z)g(z)ρidz,
(3)
ΔT=MΔPRρi.
(4)

It is evident that the uncertainty in calculated temperatures is directly proportional to ΔP and because of its inverse relationship with atmospheric density, ΔT increases as the density decreases with altitude. Also, because the seed pressure is chosen at the top of the atmosphere, a large uncertainty is expected due to the lack of routine pressure measurements at these heights compounded by the large geophysical variability that exists in the upper atmosphere due to atmospheric waves.

3. Inversion Method

The method presented here is an inversion approach. This method is applied to retrieve atmospheric temperatures from the coupled form of the system equations. The theory of mathematical inversion is based upon the relationship between the state of the system or system parameters and the observed variables of the system. This relationship can be expressed by the forward model of the system given by
F(m)=d,
(5)
where, dN×1 is the vector of observed data and mM×1 is the vector of unknown model parameters. F represents the forward model (letters in bold face represent matrices or vectors).

The inversion approach, like other optimization schemes, is an iterative approach to approximate the true value of parameters by starting from an initial value for them and updating them with an improved value, step by step by following a predefined optimization criterion. If m0 is an initial guessed value of the parameter vector m and m0+δm the true value that is sought, F(m0), becomes an estimate of d and F(m0+δm), the true value dtrue. The crux of the inversion method is to minimize the difference between the estimated and true value of m by minimizing the difference between the estimated and observed value of d. The condition on convergence of the guessed solution to the true one is implemented by minimizing the objective function (χ2), i.e., a weighted least-squares minimization (Bevington and Robinson Chap. 4 [2

2. P. R. Bevington and D. K. Robinson, Data Reduction and Error Analysis for the Physical Sciences, 2nd ed. (McGraw-Hill, 1992).

]). Following the method of maximum likelihood, the χ2 difference is defined assuming that the data are sampled from a Gaussian distribution and is expressed as follows:
χ2(X)=(XX¯)TVar1(XX¯).
(6)
This objective function can be expressed in the following expanded form:
χ2(x)=1Nj=1N(xjxj¯)2σj2.
(7)

The objective function is defined for a data vector X of size N and mean vector X¯, where xj is an element of the data vector, xj¯ is the expected value of that variable, and σj2 is the data variance for the jth variable.

A reasonable choice for the convergence condition is that the change in χ2 per degree of freedom (or for each parameter) per minimization step is less than or equal to 0.1%. If the parameters are correlated to one other, the true definition of the χ2 function for correlated data can be used, which is
χ2=(XexpXmodel)TCov1(XexpXmodel),
(8)
where Cov is the data variance-covariance matrix whose entries are the average values of the product (xixi¯)(xjxj¯), where xi and xj can be any two variables of the total X number of variables. For a system of X variables each having a sample size of M, a detailed expression for the (i,j)th term of the variance-covariance matrix can be written as
Cov(xi,xj)=k=1M(xi,kxi¯)(xj,kxj¯)M.
(9)

4. Implementation of the Inversion Approach

In the CH method, the decoupled system equations are used to obtain an approximate analytical form. The more sophisticated forward model in the inversion approach allows the assumption of constant temperature between layers to be removed. This removal is achieved by integrating the equation by using a trapezoidal-rule integration instead of a simple rectangular integration in the following manner:
z0zidPP(z)=MRz0zig(z)T(z)dz,
(10)
P(zi)=P0exp(MRz0zig(z)T(z)dz),
(11)
where the subscript “0” refers to the value of the variable at the seed altitude, z0. Using the ideal gas law, this equation can be written as
ρ(zi)RT(zi)M=P0exp(MRz0zig(z)T(z)dz).
(12)
The lidar equation can be written in the form
ρ(zi)=C(N(zi)B(zi))zi2.
(13)
In this form of the lidar equation, the origin of the coordinate system is assumed to be at the elevation of the lidar, as opposed to mean sea level. The density is proportional to the measured photocounts, N(zi), which are corrected for background, B(zi), and area. The constant of proportionality, C, depends on the lidar’s receiver and transmitter properties, as well as the atmospheric transmission, which is the same in both upward and downward directions for elastic scattering [4

4. V. A. Kovalev and W. E. Eichinger, Elastic Lidar: Theory, Practice and Analysis Method (Wiley, 2004).

]. For visible light above the heights of the aerosol layer (e.g., 25–30 km), the atmospheric transmission is essentially constant with height; hence C is height-independent.

From Eq. (13), ρ(zi) can be written in terms of the lidar return signal at the altitude zi, which modifies Eq. (12) to
N(zi)=P0MCRzi2T(zi)exp(MRz0zig(z)T(z)dz).
(14)
This nonlinear relation between N and T is the required forward model for the inversion problem. Note that it is of the form d=F(m), where the unknown model parameters, m, can be replaced by T and the observables, d, replaced with N.

A seed temperature value is expected to introduce the same amount of systematic uncertainty, as does the combination of seed pressure and seed density if both seed pressure and density are obtained from a model, which uses the ideal gas law to calculate temperature, pressure, or density from measured values of the other two quantities. Such a situation exists for the current Purple Crow Lidar (PCL) system; thus, it will not matter which of the two equations, Eq. (14) or Eq. (15), is used for the retrieval. The temperature retrievals shown in the following sections use Eq. (14) as a forward model.

The PCL’s Rayleigh-scatter system, as configured for this study, collects collects 1200 laser shots per minute in 24 m altitude range bins to form single minute count profiles throughout the night [7

7. R. J. Sica, S. Sargoytchev, P. S. Argall, E. F. Borra, L. Girard, C. T. Sparrow, and S. Flatt, “Lidar measurements taken with a large-aperture liquid mirror. 1. Rayleigh-scatter system,” Appl. Opt. 34, 6925–6936 (1995). [CrossRef]

]. For a nightly average temperature profile, the single minute scans are co-added together. To obtain a value of the normalization constant C, the nightly co-added profile, after background correction, is scaled to a density model. In this study the CIRA 1990 atmospheric model [8

8. E. L. Fleming, S. Chandra, J. J. Barnett, and M. Corney, “Zonal mean temperature, pressure, zonal wind and geopotential height as functions of latitude,” Adv. Space Res. 12, 11–59 (1990).

] was used as a standard density model. Normalization of the density profiles is done by integrating the model and measurements between 45 and 60 km. The covariance matrix, as previously described, is constructed from the night’s single minute scans and then used with the inversion approach on the nightly co-added measurements to retrieve a single average temperature profile for a night.

In this study the seed pressure value P0 was also obtained from the CIRA model [8

8. E. L. Fleming, S. Chandra, J. J. Barnett, and M. Corney, “Zonal mean temperature, pressure, zonal wind and geopotential height as functions of latitude,” Adv. Space Res. 12, 11–59 (1990).

]. However since the starting altitude is in the range of radiosonde measurements(35km), a radiosonde-derived density can be used if available for a more accurate seed pressure.

All of the temperatures retrieved for this study were spatially and temporally co-added to get nightly averaged temperature profiles. Co-adding counts increases the SNR, which helps extend the upper altitude limit of integration. Co-adding was done over a 500 m vertical resolution. In the CH method, using PCL data, it is customary to start integration, after co-adding, from the altitude where the SNR becomes approximately equal to 2 to separate data from noise [9

9. P. S. Argall and R. J. Sica, “A comparison of Rayleigh and sodium lidar temperature climatologies,” Ann. Geophys. 25, 27–35 (2007). [CrossRef]

].

5. Error Analysis

Over a night’s measurements the distribution of total uncertainties for single-minute lidar measurements tend to a Gaussian distribution for similar geophysical conditions. The assumption of a Gaussian distribution is adequate here for the random uncertainties as the mean count rate even at the greater heights is sufficiently large that a Poisson distribution is indistinguishable from a Gaussian distribution [2

2. P. R. Bevington and D. K. Robinson, Data Reduction and Error Analysis for the Physical Sciences, 2nd ed. (McGraw-Hill, 1992).

]. This condition provides a means to model the uncertainty budget by using a Gaussian random number generator and then multiplying the output by the uncertainty in the converged upon lidar returned count profile Nmodel.

Similar steps can be taken for other systematic uncertainties applied in order to obtain an accurate estimate for the uncertainty of Nmodel for uncorrelated uncertainties. With the necessary corrections to the variance completed, the noisy count profiles needed for the Monte Carlo error analysis can be generated by multiplying the error at a given altitude by a random normalized Gaussian distribution generator and adding this quantity to Nmodel such that
NMC(z)=N(z)+σN(z)λ,
(17)
where NMC is the generated Monte Carlo count profile, and λ is the normalized random Gaussian number generator. If a large number of noisy counts profiles are generated by adding Gaussian noise to the counts profile using the aforementioned method, the known analytic uncertainty and the standard deviation from all the Monte Carlo generated profiles can be compared. It was found that the reported uncertainties for both were in agreement for the entire profile. In addition, with the developed Monte Carlo method, it is possible to isolate systematic and random uncertainties and model the effect on the final uncertainty in the temperature profile. For the PCL, it was found that 50 runs were sufficient for the Monte Carlo error analysis to converge.

6. Results

The primary disadvantage of the CH method is that the top 10–15 km of the temperature retrieval has to be discarded due to the uncertainty associated with the choice of seed pressure, which propagates downward. With the inversion approach, a bottom seed pressure can be used, and due to the numerical optimization technique, an incorrect choice of seed pressure does not impact the retrieved temperature profile as much as the CH method. This assertion was confirmed by taking some representative nights of PCL measurements, then applying both methods and changing the seed pressure used by equal proportions and viewing the effect on the retrieved temperature profiles. The temperature profile retrieved with the initial seed pressure without any variations can be taken as the true profile for the night, while the other profiles model incorrect choices of pressure and the effect on the retrieved temperatures. Figures 1 and 2 show the results for the CH method and the inversion approach respectively for the night of 1 September 2005. The initial seed pressure used in each case was 0.00021 hPa for the top-down CH method, and 2.92 hPa for the bottom-up inversion approach. For both cases the altitude bins were co-added in bins of 504 m resolution, and then smoothed with a low-pass 3s and 5s filter [12

12. R. W. Hamming, Digital Filters (Prentice-Hall, 1977).

].

Fig. 1. Temperature retrievals using the CH method for the Rayleigh-scatter lidar measurements on 1 September 2005. These retrievals show the effect of varying the choice of the top seed pressure. The right panel shows the temperature differences from the profile retrieved using the original seed pressures given in the text.
Fig. 2. Temperature retrievals using an inversion approach for PCL Rayleigh-scatter lidar measurements on 1 September 2005. These retrievals show the effect of varying the choice of the top seed pressure. The right panel shows the temperature differences from the profile retrieved using the original seed pressures given in the text.

Fig. 3. Comparison of the statistical uncertainties reported for the night of 1 September 2005 for the CH method and an inversion approach from Figs. 1 and 2 respectively with no variation in seed pressure. The uncertainties with the Grid Search method were found using the Monte Carlo approach described in Section 5.

Fig. 4. Temperature retrieval from an inversion approach for August 2000. There were a total of 270 one-minute scans for this night. The vertical resolution is 500 m. The red portion of the plot indicates the extra temperature measurements gained when compared to the CH method where the top 10–15 km would have an unacceptably high systematic uncertainty.
Fig. 5. Temperature retrieval from an inversion approach for 22 August 1995, using 483 one-minute scans, in the same format as Fig. 4.
Fig. 6. Temperature retrieval from an inversion approach for 5 May 2006, using 385 one-minute scans, in the same format as Fig. 4.

7. Conclusions

Using mathematical inversion techniques allows Rayleigh-scatter temperature profiles retrieved from lidar measurements to be extended up to heights where previously retrievals were not possible due to the limitations in the CH method in regard to the top-down pressure integration. Gaining 10 km at the top of a lidar temperature profile requires increasing the power-aperture product of the lidar by a factor of 4. For systems such as the lidars in the Network for the Detection of Atmospheric Climate Change (NDACC) that typically already employ high-power laser transmitters, this change would require doubling the telescope diameter, which is typically prohibitively expensive. Hence, applying an inversion approach is equivalent to a significant hardware upgrade to an existing lidar, with the added benefit that past as well as future measurements enjoy the increase in effective power-aperture product.

Implementing an inversion approach for temperature retrievals offers two significant advantages compared to the CH method. First, the bottom upwards integration allows the choice of a seed pressure in the stratosphere rather than the mesosphere or the lower thermosphere. The geophysical variability at all spatial-temporal scales is much smaller in the stratosphere than the upper atmosphere. Hence, the systematic uncertainty due to the choice of seed pressure is much smaller. In addition, an inversion approach allows this uncertainty to be propagated over all heights in the model, contributing less to the systematic uncertainty near the top of the retrieval compared to the CH method.

Second, an inversion approach allows for a more thorough, albeit more complicated, uncertainty analysis. The use of a Monte Carlo technique allows systematic as well as random uncertainties to be estimated.

We would like to acknowledge support of this research by the Natural Sciences and Engineering Research Council of Canada. We would also like to thank the reviewers for their insightful and helpful comments.

References

1.

A. Hauchecorne and M. L. Chanin, “Density and temperature profiles obtained by lidar between 35 and 70 km,” Geophys. Res. Lett. 7, 565–568 (1980). [CrossRef]

2.

P. R. Bevington and D. K. Robinson, Data Reduction and Error Analysis for the Physical Sciences, 2nd ed. (McGraw-Hill, 1992).

3.

P. C. Mahalanobis, “On the generalized distance in statistics,” Proc. Natl. Inst. Sci. India 2, 49–55 (1936).

4.

V. A. Kovalev and W. E. Eichinger, Elastic Lidar: Theory, Practice and Analysis Method (Wiley, 2004).

5.

P. B. Russell and B. M. Morley, “Orbiting lidar simulations. 2: density, temperature, aerosol, and cloud measurements by a wavelength combining technique,” Appl. Opt. 21, 1554–1563 (1982). [CrossRef]

6.

J. P. Thayer, N. B. Nielsen, R. E. Warren, C. J. Heinselman, and J. Sohn, “Rayleigh lidar system for middle atmosphere research in the arctic,” Opt. Eng. 36, 2045–2061 (1997). [CrossRef]

7.

R. J. Sica, S. Sargoytchev, P. S. Argall, E. F. Borra, L. Girard, C. T. Sparrow, and S. Flatt, “Lidar measurements taken with a large-aperture liquid mirror. 1. Rayleigh-scatter system,” Appl. Opt. 34, 6925–6936 (1995). [CrossRef]

8.

E. L. Fleming, S. Chandra, J. J. Barnett, and M. Corney, “Zonal mean temperature, pressure, zonal wind and geopotential height as functions of latitude,” Adv. Space Res. 12, 11–59 (1990).

9.

P. S. Argall and R. J. Sica, “A comparison of Rayleigh and sodium lidar temperature climatologies,” Ann. Geophys. 25, 27–35 (2007). [CrossRef]

10.

JCGM, “Evaluation of measurement data: guide to the expression of uncertainty in measurement,” Tech. Rep. (Joint Committee for Guides in Meteorology, 2008).

11.

JCGM, “Evaluation of measurement data,” supplement 1 to the “Guide to the expression of uncertainty in measurement propagation of distributions using a Monte Marlo method,” Tech. Rep. (Joint Committee for Guides in Meteorology, 2008).

12.

R. W. Hamming, Digital Filters (Prentice-Hall, 1977).

OCIS Codes
(000.2170) General : Equipment and techniques
(010.3640) Atmospheric and oceanic optics : Lidar
(280.3640) Remote sensing and sensors : Lidar
(280.6780) Remote sensing and sensors : Temperature

ToC Category:
Atmospheric and Oceanic Optics

History
Original Manuscript: July 10, 2012
Revised Manuscript: October 11, 2012
Manuscript Accepted: October 17, 2012
Published: November 19, 2012

Citation
Jaya Khanna, Justin Bandoro, R. J. Sica, and C. Thomas McElroy, "New technique for retrieval of atmospheric temperature profiles from Rayleigh-scatter lidar measurements using nonlinear inversion," Appl. Opt. 51, 7945-7952 (2012)
http://www.opticsinfobase.org/ao/abstract.cfm?URI=ao-51-33-7945


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References

  1. A. Hauchecorne and M. L. Chanin, “Density and temperature profiles obtained by lidar between 35 and 70 km,” Geophys. Res. Lett. 7, 565–568 (1980). [CrossRef]
  2. P. R. Bevington and D. K. Robinson, Data Reduction and Error Analysis for the Physical Sciences, 2nd ed. (McGraw-Hill, 1992).
  3. P. C. Mahalanobis, “On the generalized distance in statistics,” Proc. Natl. Inst. Sci. India 2, 49–55 (1936).
  4. V. A. Kovalev and W. E. Eichinger, Elastic Lidar: Theory, Practice and Analysis Method (Wiley, 2004).
  5. P. B. Russell and B. M. Morley, “Orbiting lidar simulations. 2: density, temperature, aerosol, and cloud measurements by a wavelength combining technique,” Appl. Opt. 21, 1554–1563 (1982). [CrossRef]
  6. J. P. Thayer, N. B. Nielsen, R. E. Warren, C. J. Heinselman, and J. Sohn, “Rayleigh lidar system for middle atmosphere research in the arctic,” Opt. Eng. 36, 2045–2061 (1997). [CrossRef]
  7. R. J. Sica, S. Sargoytchev, P. S. Argall, E. F. Borra, L. Girard, C. T. Sparrow, and S. Flatt, “Lidar measurements taken with a large-aperture liquid mirror. 1. Rayleigh-scatter system,” Appl. Opt. 34, 6925–6936 (1995). [CrossRef]
  8. E. L. Fleming, S. Chandra, J. J. Barnett, and M. Corney, “Zonal mean temperature, pressure, zonal wind and geopotential height as functions of latitude,” Adv. Space Res. 12, 11–59 (1990).
  9. P. S. Argall and R. J. Sica, “A comparison of Rayleigh and sodium lidar temperature climatologies,” Ann. Geophys. 25, 27–35 (2007). [CrossRef]
  10. JCGM, “Evaluation of measurement data: guide to the expression of uncertainty in measurement,” Tech. Rep. (Joint Committee for Guides in Meteorology, 2008).
  11. JCGM, “Evaluation of measurement data,” supplement 1 to the “Guide to the expression of uncertainty in measurement propagation of distributions using a Monte Marlo method,” Tech. Rep. (Joint Committee for Guides in Meteorology, 2008).
  12. R. W. Hamming, Digital Filters (Prentice-Hall, 1977).

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