Geneva, September 2010 EARLINET-ASOS Symposium Second GALION Workshop Uncertainties evaluation for aerosol optical properties Aldo Amodeo CNR-IMAA Uncertainties evaluation for aerosol optical properties Aldo Amodeo CNR-IMAA

Geneva, September 2010 EARLINET-ASOS Symposium Second GALION Workshop OUTLINE  General concepts  Source of errors in lidar measurements  The problem of the calculation of the statistical error  The problem of the calculation of the sistematic error

Geneva, September 2010 EARLINET-ASOS Symposium Second GALION Workshop Basic concepts Lidar measurements, as for all the measurements, need the estimation of the associated error, because every measurement without error could have 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.

Geneva, September 2010 EARLINET-ASOS Symposium Second GALION Workshop a)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); b)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 Z ovl ) are caused by the overlap function (Wandinger and Ansmann, 2002). SOURCES OF ERRORS

Geneva, September 2010 EARLINET-ASOS Symposium Second GALION Workshop Error calculation Parameter Raw lidar signal Extinction coefficient Backscatter coefficient Acquisition technique Photoncounting Analog

Geneva, September 2010 EARLINET-ASOS Symposium Second GALION Workshop 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 X 

Geneva, September 2010 EARLINET-ASOS Symposium Second GALION Workshop 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:  : expected mean number of events within the time T: Standard deviation of the observed number : = 1 = 4 = 10 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. When  is large, the Poisson distribution P  ( ) is well approximeted by Gauss distribution with the same mean and standard deviation:

Geneva, September 2010 EARLINET-ASOS Symposium Second GALION Workshop 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 Statistical error

Geneva, September 2010 EARLINET-ASOS Symposium Second GALION Workshop 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.

Geneva, September 2010 EARLINET-ASOS Symposium Second GALION Workshop

Geneva, September 2010 EARLINET-ASOS Symposium Second GALION Workshop

Geneva, September 2010 EARLINET-ASOS Symposium Second GALION Workshop 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.

Geneva, September 2010 EARLINET-ASOS Symposium Second GALION Workshop Signal data binning 8 points

Geneva, September 2010 EARLINET-ASOS Symposium Second GALION Workshop Signal data binning

Geneva, September 2010 EARLINET-ASOS Symposium Second GALION Workshop Signal data binning

Geneva, September 2010 EARLINET-ASOS Symposium Second GALION Workshop 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)

Geneva, September 2010 EARLINET-ASOS Symposium Second GALION Workshop

Geneva, September 2010 EARLINET-ASOS Symposium Second GALION Workshop

Geneva, September 2010 EARLINET-ASOS Symposium Second GALION Workshop

Geneva, September 2010 EARLINET-ASOS Symposium Second GALION Workshop

Geneva, September 2010 EARLINET-ASOS Symposium Second GALION Workshop

Geneva, September 2010 EARLINET-ASOS Symposium Second GALION Workshop Error comparison with the background subtraction Where  Background is the standard deviation of the calculated background.

Geneva, September 2010 EARLINET-ASOS Symposium Second GALION Workshop

Geneva, September 2010 EARLINET-ASOS Symposium Second GALION Workshop Error calculation Parameter Raw lidar signal Extinction coefficient Backscatter coefficient Acquisition technique Photoncounting Analog

Geneva, September 2010 EARLINET-ASOS Symposium Second GALION Workshop ERROR CALCULATION IN THE AEROSOL EXTINCTION COEFFICIENT RETRIEVING The aerosol extinction coefficient,  aer  can be determined from the N 2 (or O 2 ) Raman backscattering signals through the application of the expression (Ansmann et al., 1990; Ansmann et al., 1992): P(z)  power received from distance z  at the Raman wavelength   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.

Geneva, September 2010 EARLINET-ASOS Symposium Second GALION Workshop HOW TO CALCULATE THE ERROR? Analytical or Numerical techniques? PROBLEMS:  difficulty in error propagation when handling procedures, such as signal smoothing, are applied. ANALYTICAL NUMERICAL (Montecarlo techniques) ADVANTAGES:  no difficulty related to error propagation calculation, whatever signal handling procedure is used. 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. 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.

Geneva, September 2010 EARLINET-ASOS Symposium Second GALION Workshop  1.  b  b+1..  k Extracted signals Solutions from extracted signals Solution with errors equal to the deviations from the solutions obtained from the extracted signals. Measured signal Solution Numerical technique

Geneva, September 2010 EARLINET-ASOS Symposium Second GALION Workshop 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 u 1 and u 2 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 n k =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.

Geneva, September 2010 EARLINET-ASOS Symposium Second GALION Workshop 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:

Geneva, September 2010 EARLINET-ASOS Symposium Second GALION Workshop EXTINCTION ERROR DEPENDENCE ON THE USED ALGORITHM (Calculated by Montecarlo technique)

Geneva, September 2010 EARLINET-ASOS Symposium Second GALION Workshop 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

Geneva, September 2010 EARLINET-ASOS Symposium Second GALION Workshop 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 z 0 that is aerosol free:

Geneva, September 2010 EARLINET-ASOS Symposium Second GALION Workshop CONTRIBUTIONS TO THE BACKSCATTER STATISTICAL ERROR

Geneva, September 2010 EARLINET-ASOS Symposium Second GALION Workshop Error calculation Parameter Raw lidar signal Extinction coefficient Backscatter coefficient (Raman/elastic, elastic only) Acquisition technique Photoncounting Analog

Geneva, September 2010 EARLINET-ASOS Symposium Second GALION Workshop

Geneva, September 2010 EARLINET-ASOS Symposium Second GALION Workshop

Geneva, September 2010 EARLINET-ASOS Symposium Second GALION Workshop

Geneva, September 2010 EARLINET-ASOS Symposium Second GALION Workshop

Geneva, September 2010 EARLINET-ASOS Symposium Second GALION Workshop Signal temporal averaging Binning Background subtraction Processing Error propagation: analytical numerical Background determination Example of possible a procedure of analysis and error propagation for analog signals Numerical technique could be used in the more useful step of the analysis

Geneva, September 2010 EARLINET-ASOS Symposium Second GALION Workshop Summation of the signals Binning Background subtraction Processing Error propagation: analytical numerical Background determination Example of possible a procedure of analysis and error propagation for photoncounting signals Numerical technique could be used in the more useful step of the analysis

Geneva, September 2010 EARLINET-ASOS Symposium Second GALION Workshop 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.

Geneva, September 2010 EARLINET-ASOS Symposium Second GALION Workshop 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, February 2008

Geneva, September 2010 EARLINET-ASOS Symposium Second GALION Workshop 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

Geneva, September 2010 EARLINET-ASOS Symposium Second GALION Workshop Influence of the air density profile on Raman extinction profiles ● temperature gradient → large effect ● absolute temperature → large effect

Geneva, September 2010 EARLINET-ASOS Symposium Second GALION Workshop Influence of the absolute temperature on Raman extinction profiles effect of the absolute temperature increases with height (optical depth)

Geneva, September 2010 EARLINET-ASOS Symposium Second GALION Workshop Influence of the air density profile on Raman lidar-ratio profiles

Geneva, September 2010 EARLINET-ASOS Symposium Second GALION Workshop Influence of the Angstrom exponent on Raman extinction retrievals

Geneva, September 2010 EARLINET-ASOS Symposium Second GALION Workshop Influence of the lidar-ratio assumption on Klett backscatter retrievals

Geneva, September 2010 EARLINET-ASOS Symposium Second GALION Workshop Influence of the lidar-ratio assumption on Klett backscatter retrievals largest effect at smaller wavelength

Geneva, September 2010 EARLINET-ASOS Symposium Second GALION Workshop Influence of the calibration on backscatter retrievals at different wavelengths

Geneva, September 2010 EARLINET-ASOS Symposium Second GALION Workshop Influence of the calibration on backscatter retrievals at different wavelengths largest effect at larger wavelength

Geneva, September 2010 EARLINET-ASOS Symposium Second GALION Workshop Errors due to a depolarization dependent receiver transmission

Geneva, September 2010 EARLINET-ASOS Symposium Second GALION Workshop Errors due to a depolarization dependent receiver transmission

Geneva, September 2010 EARLINET-ASOS Symposium Second GALION Workshop Errors due to a depolarization dependent receiver transmission

Geneva, September 2010 EARLINET-ASOS Symposium Second GALION Workshop Acknowledgements EARLINET-ASOS project funded by the European Commission (EC) under grant RICA