Uncertainties evaluation for aerosol optical properties Aldo Amodeo CNR-IMAA

1. Uncertainties evaluation for aerosol optical properties Aldo Amodeo CNR-IMAA

2. OUTLINE • Generalconcepts • Source oferrors in lidar measurements • The problemof the calculationof the statisticalerror • The problemof the calculationof the sistematicerror

3. Basicconcepts Lidar measurements, asforall the measurements, need the estimationof the associatederror, becauseeverymeasurementwithouterrorcouldhave no meaning. The determination of the error is not a simple task, especially when several operations are applied in the data analysis: smoothing, averaging, background subtraction, gluing, analysis algorithm. Two kinds of errors can be distinguished: statistical error and systematic error.

4. SOURCES OF ERRORS • Statistical error • mainly due to the to signal detection [background of sky and dark current of detector] (Theopold and Bösenberg, 1988); • directly related to this kind of error, there is the error introduced by operational procedures such as signal averaging during varying atmospheric extinction and scattering conditions (Ansmann et al., 1992; Bösenberg, 1998); • Systematic error • due to uncertainties related to instruments, fixed parameters in the retrieval,… • the systematic error associated with the estimate of temperature and pressure profiles (Ansmann et al., 1992); • the systematic error associated with the estimate of the ozone profiles in the UV (Ansmann et al., 1992); • the systematic error associated with the wavelength dependence parameter k (Ansmann et al., 1992; Whiteman, 2000); • the systematic error associated with the multiple scattering (Ansmann et al., 1992; Wandinger, 1998; Whiteman, 2000); • extinction uncertainties (up to 50% for heights below Zovl) are caused by the overlap function (Wandinger and Ansmann, 2002).

5. Errorcalculation • Parameter • Raw lidar signal • Extinctioncoefficient • Backscattercoefficient • Acquisitiontechnique • Photoncounting • Analog

6.  X Gaussian distribution (suitable for analog mode) If the variable can assume in principle continuous values, if a measurement is affected by many source of random errors, and systematic errors are negligible, the measured values will be distributed according a bell curve, centred on the true value of x. If measurements are affected by not negligible systematic effects, the distribution of the measurements will not centred around the true value. Variables affected only by statistical errors are described by Gaussian (or normal) distribution: X: true value of x, centre of the distribution, mean value after many measurements. : distribution width standard deviation after many measurements

7.  = 1   = 4   = 10 Poisson distribution (suitable for photoncounting mode) The Poisson distribution describes experiments in which are counted events that happen randomly, but with a defined mean rate. The variable is discrete. If we count during a time interval T, the probability to observe  events is given by the Poisson function: The horizontal axis is the index . The function is defined only at integer values of . The connecting lines are only guides for the eye and do not indicate continuity. : expected mean number of events within the time T: Standard deviation of the observed number : When  is large, the Poisson distribution P() is well approximeted by Gauss distribution with the same mean and standard deviation:

8. Statisticalerror • This kind of information is contained in the measured standard deviation (z) of the lidar signal. • Possible techniques of evaluation • Analytical • Numerical (Montecarlo) • Calculation of the standard deviation among the single solutions

9. Error calculation • Parameter • Raw lidar signal • Extinction coefficient • Backscatter coefficient • Acquisition technique • Photoncounting • Analog • Square root of the counts. • The error on the subtracted background raw signal should include the propagation of the error on the background. • where n is the number of bins used to calculate the background.

12. Signal data binning 4 points • Each point is obtained by: • summing the counts contained in a certain number of bins • associating as height the mean of the height range relative to the binned points.

13. Signal data binning 8 points

14. Signal data binning

15. Signal data binning

16. Error calculation • Parameter • Raw lidar signal • Extinction coefficient • Backscatter coefficient • Acquisition technique • Photoncounting • Analog • Standard deviation calculated on the averaging time interval. • The error on the subtracted background raw signal should include the propagation of the error on the background (sky and electronic)

22. Error comparison with the background subtraction Where Background is the standard deviation of the calculated background.

24. Error calculation • Parameter • Raw lidar signal • Extinction coefficient • Backscatter coefficient • Acquisition technique • Photoncounting • Analog

25. ERROR CALCULATION IN THE AEROSOL EXTINCTION COEFFICIENT RETRIEVING The aerosol extinction coefficient, aer, can be determined from the N2 (or O2) Raman backscattering signals through the application of the expression (Ansmann et al., 1990; Ansmann et al., 1992): P(z): power received from distance zat the Raman wavelength l0: transmitted laser wavelength N(z): atmospheric number density mol,: extinction coefficient due to absorption and Rayleigh scattering by atmospheric gases, and where particle scattering is assumed to be proportional to -1.

26. ANALYTICAL HOW TO CALCULATE THE ERROR? Analytical or Numerical techniques? NUMERICAL (Montecarlo techniques) • ADVANTAGES: • no difficulty related to error propagation calculation, whatever signal handling procedure is used. It is important to know the signal standard deviation for each height and the type of distribution function. In the case of photon-counting, this is a Poisson distribution. • PROBLEMS: • difficulty in error propagation when handling procedures, such as signal smoothing, are applied. THE PRINCIPLE This procedure is based on a random extraction of new lidar signals, each bin of which is considered as a sample element of a given probability distribution with the experimentally observed mean value and standard deviation. The extracted lidar signals are then processed to retrieve a set of solutions from which the standard deviation as a function of the height is estimated.

27. 1 . . b b+1. . k Measured signal Solution Solution with errors equal to the deviations from the solutions obtained from the extracted signals. Numerical technique Extracted signals Solutions from extracted signals

28. Random extractors • Several procedures exist. Generally they start from the random generation of numbers according to the uniform distribution and transform the extracted numbers in numbers following the desired distribution. • Examples of simple algorithms for some extractors: • Gaussian distribution • If u1 and u2 are uniform on (0,1), then • are independent and Gaussian distributed with mean 0 and =1 • Poisson distribution • Iterate until a successful is made: • begin with k=1 and set A=1 to start • generate u uniform in (0,1) • replace A with uA • if now A<exp(-) where  is the Poisson parameter, accept nk=k-1 and stop; • otherwise increment k by 1, generate a new u and repeat, always starting with the value of A left from the previous try.

29. COMPARISON BETWEEN EXTINCTION ERROR CALCULATED BY ANALITYC AN MONTECARLO TECHNIQUES Extinction calculated by SLIDING LINEAR FIT so that it is simple to calculate the derivative in the formula. The ANALYTICAL error is:

30. EXTINCTION ERROR DEPENDENCE ON THE USED ALGORITHM (Calculated by Montecarlo technique)

31. Error calculation • Parameter • Raw lidar signal • Extinction coefficient • Backscatter coefficient (Raman/elastic, elastic only) • Acquisition technique • Photoncounting • Analog • Technique: Analytical • Numerical (Montecarlo) • Calculation of the standard deviation among the single solutions

32. ERROR CALCULATION IN THE AEROSOL BACKSCATTER COEFFICIENT RETRIEVING The aerosol backscatter coefficient, aer, can be determined from the ratio between the two lidar signals at laser and Raman wavelengths L and R(Ansmann et al., 1992): The constant C* includes instrumental and geometrical system properties and is retrieved by normalizing lidar signal at a reference height z0 that is aerosol free:

33. CONTRIBUTIONS TO THE BACKSCATTER STATISTICAL ERROR

34. Error calculation • Parameter • Raw lidar signal • Extinction coefficient • Backscatter coefficient (Raman/elastic, elastic only) • Acquisition technique • Photoncounting • Analog

39. Example of possible a procedure of analysis and error propagation for analog signals Signal temporal averaging Binning Background determination Background subtraction Numerical technique could be used in the more useful step of the analysis Processing Error propagation: analytical numerical

40. Example of possible a procedure of analysis and error propagation for photoncounting signals Summation of the signals Binning Background determination Background subtraction Processing Numerical technique could be used in the more useful step of the analysis Error propagation: analytical numerical

41. If you use other techniques (smoothing, merging or other) or apply procedures in different order take care to apply the right error propagation procedure. For example, in the case of merging, take into account the product for the normalization of the two signals and also the possible difference in typology between the two signals: analog and digital.

42. Some systematic errors ● Influence of the air-density profile ● Influence of the Angstrom-exponent parameter ● Influence of the lidar ratio assumption for the Klett retrievals at different wavelengths ● Influence of the calibration on backscatter retrievals at different wavelengths ● Errors due to a depolarization dependent receiver transmission from EARLINET ASOS training course for the retrieval of optical aerosol properties (Ina Mattis) Thessaloniki, 25-26 February 2008

43. Influence of the air density profile ● temperature gradient → small effect ● absolute temperature → larger effect β par ~ SigRatio − β mol β mol ~ number density of air molecules ~ T

44. Influence of the air density profile on Raman extinction profiles ● temperature gradient → large effect ● absolute temperature → large effect

45. Influence of the absolute temperature on Raman extinction profiles effect of the absolute temperature increases with height (optical depth)

46. Influence of the air density profile on Raman lidar-ratio profiles

47. Influence of the Angstrom exponent on Raman extinction retrievals

48. Influence of the lidar-ratio assumption on Klett backscatter retrievals

49. Influence of the lidar-ratio assumption on Klett backscatter retrievals largest effect at smaller wavelength

50. Influence of the calibration on backscatter retrievals at different wavelengths