1. Comments about integral parameters of atmospheric aerosol particle size distribution and two power law Comments about integral parameters of atmospheric aerosol particle size distribution and two power law Hannes.Tammet@ut.ee with acknowledgements to Markku Kulmala and colleagues from University of Helsinki providing the measurements -

2. Introduction A free dataset of atmospheric aerosol size distribution measurements Hyytiala08_10aerosol can be downloaded from web, see: http://dx.doi.org/10.15155/repo-3 The data is described in the paper Tammet, H., Kulmala, M. (2014). Empiric equations of coagulation sink of fine nanoparticles on background aerosol optimized for boreal zone. Boreal Environ. Res., 19, 115–126. An example of previous usage of the dataset is Tammet, H., Kulmala, M. (2014). Performance of four-parameter analytical models of atmospheric aerosol particle size distribution. J. Aerosol Sci., 77, 145–157. I am going to present some fragmentary examples how this dataset could be used to study different problems in atmospheric aerosol research.

3. Size distribution  is a smooth continuous function  defined from d p = 0 to d p = ∞. Theory: or  include limited number of fractions,  available in limited boundaries. Measurements: ~ + noise ? measured theoretical fraction fraction

4. Weller, R., Schmidt, K., Teinilä, K., Hillamo, R. (2015) Natural new particle formation at the coastal Antarctic site Neumayer. Atmos. Chem. Phys. Discuss., 15, 15655-15681. Data available http://doi.pangaea.de/10.1594/PANGAEA.845000 EXCERPT: Hour 6.36.77.27.88.49.09.710111231415161718 nm 110000000000001008816382 1200000013800000531814169 130002350001060836658100943922 1400002040002560000918069 15000000009083715710401171095 1600000000970005306793 170000000115900711195008071 1846300018401261611530000444148 1900000000000010706779 2000000001061950000444137 21000000000005750444169 22040600007016700011950339188 23000000000001760448373 Find “noise”  0 hits Find “fluctuation”  0 Diameter

5. Hour/nm2.93.43.94.55.26.07.08.1μm==>56789 11000536112689……2130000 1200031862061……00000 130072662523641……3760000 14008211083474……2943253100 1598400744303351……213 000 1600133530 4462……7010000 17008202538293……11394893760212 18021578225385686……814213000 194936293976126186245……11072133262130 2000738128185301497……1139375000 210032396185230325……814294000 22016252336113209354……1627407489327214 23000850142194330……130210586512940 Tammet, H., Kulmala, M. (2014) Empiric equations of coagulation sink of fine nanoparticles on background aerosol optimized for boreal zone. Boreal Environ. Res., 19, 115–126. Dataset Hyytiala08_10aerosol ===> http://dx.doi.org/10.15155/repo-3 EXCERPT: Multiplied with 10 6

6. Integrals, moments, concentrations Integrals of order q : NB: Largest airborne particle d p = 12756 km! Moments of order q : Number concentration N = M 0 ? 2.7 × 10 19 cm –3 ??? Mass concentration C m = ρC V = (π/ 6) ρM 3 ???  PM1 = (π/6) ρ C 3 (3 nm, 1 μm)  PM2.5 = (π/6) ρ C 3 (3 nm, 2.5 μm)  PM10 = (π/6) ρ C 3 (3 nm, 10 μm) Commonly accepted integrals have definite upper limits Why 3 nm? How to choose and justify the values of limits?

7. Average size distributions, cm -3

8. Proposal  PN3 = C 0 (3 nm, 10 μm),  PN5 = C 0 (5 nm, 10 μm),  PN7.5 = C 0 (7.5 nm, 10 μm),  PV1 = (π/6)C 3 (3 nm, 1 μm),  PV2.5 = (π/6)C 3 (3 nm, 2.5 μm),  PV10 = (π/6)C 3 (3 nm, 10 μm). Choose standard limiting diameters and define standard integrals (standard upper limits are already commonly accepted as 1 μm, 2.5 μm, and 10 μm). Standard lower limits could be 3, 5, and 7.5 nm ? Standard integrals: ( PM1 = ρPV1, PM1 = ρPV1, PM10 = ρPV10 )

9. Now we try to learn: At first a special case: we know that PV10 > PV1 and look for a quantitative description of relative difference between PV1 and PV10. A measure  effect of upper limit d 2 in C 3 ( 3 nm, d 2 ).  effect of lower limit d 1 in C 0 (d 1, 10 μm), can be called the deficiency of PV1 in relation to PV10.

10. Some general agreements: The results will be easier to survey when we agree at first the standard wide range of diameters. Following calculations are made on the assumption that the limits of the wide range (min, max) are min = 3 nm, max = 10 μm. Deficiencies in relation to the wide range integrals are:  Lower deficiency Δ q (min, x) = 100% × C q (min, x) / C q (min, max).  Upper deficiency Δ q (x, max) = 100% × C q (x, max) / C q (min, max).

11. The deficiencies are specific of the size distribution. We have dataset Hyytiala08_10aerosol and can easily examine the deficiencies for every one-hour distribution as well as for the average size distribution. Introductory calculations show that statistical distribution of deficiencies in the sample of 21682 one-hour measurements is asymmetric and has a high excess. The average of deficiencies does not equal the deficiency estimated for the long-time average size distribution. Thus we should learn the statistical distribution of deficiencies through the dataset of 21682 measurements.

12. Upper deficiency for q = 3

13. Lower deficiency for q = 0

14. Upper deficiency for q = 0

15. Lower deficiency for q = 1

16. Upper deficiency for q = 2

17. Interim conclusions Median deficiencies are about as expected. Higher percentiles are very large. Why? Effect of noise? This is still unsolved problem. How to suppress the effect of noise?  Averages over longer time period (3 hours, 24 hours…)  Smoothing of dN / d(lg d p ) curve

18. Hour/nm2.93.43.94.55.26.07.08.1μm==>56789 11000536112689……2130000 1200031862061……00000 130072662523641……3760000 14008211083474……2943253100 1598400744303351……213 000 1600133530 4462……7010000 17008202538293……11394893760212 18021578225385686……814213000 194936293976126186245……11072133262130 2000738128185301497……1139375000 210032396185230325……814294000 22016252336113209354……1627407489327214 23000850142194330……130210586512940 Tammet, H., Kulmala, M. (2014) Empiric equations of coagulation sink of fine nanoparticles on background aerosol optimized for boreal zone. Boreal Environ. Res., 19, 115–126. Dataset Hyytiala08_10aerosol ===> http://dx.doi.org/10.15155/repo-3 EXCERPT: Multiplied with 10 6

19. How to suppress the effect of noise?  Averages over longer time period (3 hours, 24 hours…)  Smoothing of dN / d(lg d p ) curve:  Method of Junge,  Method of Deirmendjian,  Two-power model:

20. Two-power model Junge Deirmendjian

21. A sample diagram of one-hour measurement Two-power approximation Measurement

22. Deviation of integrals caused by smoothing (or caused by noise in unsmoothed data?) x = integral according to the 60-fraction measurements y = integral according to the two-power approximation deviation = ln ( y / x ) × 100% Preliminary results through 21682 measurements (Hyytiala08_10aerosol) IntegralPN3PN5PN7.5PV1PV2.5PV10 Average deviation, %6.40.7– 3.3– 9.4– 11.1– 3.7 Std of deviation, %26 11 928

23. We see considerable systematic deviation of integrals estimated immediately according to measurements and estimated according to parameters of two-power model. However, the two-power parameters in the dataset Hyytiala08_10aerosol are claimed to be adjusted so that the systematic shift should be zero. How to explain this observation?

24. Preliminary explanation The parameters of two-power approximation are adjusted with aim to suppress systematic relative shift of distribution function uniformly in the full size range: Most of an integral is accumulated in a limited size range where the adjustment of two-power parameters is not optimized. The deviations on the wings of the distribution curve have here the same weight as the deviations in the central region. The systematic shift of integrals is a different quantity:

25. A sample diagram B A Two-power approximation Measurement

26. An extra question Quite often a measurement should provide only few integral parameters of aerosol. Why to measure 60 fractions? Develop a two-power aerosol size spectrometer where smoothing is performed in the instrument. It should have as a minimum of 4 measurement channels. Some redundancy (e.g. 8 channels) would improve the reliability. A proposal

27. A possible approach (and maybe a good business plan):  Create a mathematical model of the spectrometer as a program code, which delivers the values of two-power parameters a, b, p, and d 0 according to a 60-fraction size distribution picked from the dataset Hyytiala08_10aerosol. Do not forget add some random noise to the spectrometer channel signals before estimating the two-power parameters.  Test the spectrometer model processing all 21682 measurements and comparing the values of two-power parameters delivered by the model with the tabulated values of the same parameters available for all measurements in additional columns of the Hyytiala08_10aerosol dataset.  Vary the technical parameters of the spectrometer with aim to achieve best fit of model-calculated two-power parameters to the tabulated two-power parameters.  If the results are satisfactory, then design and test a real instrument. Otherwise try to modify the structure of the model instrument and optimize again the values of technical parameters. A spectrometer has definite discrete structure described with number of electrodes etc., and continuous technical parameters (dimensions, voltages etc.). How to optimize a two-power spectrometer? An important application of the dataset

28. Project IPSD IDENTIFYIER OF PARTICLE SIZE DISTRIBUTION Background The output of a standard aerosol spectrometer is the multifraction presentation of the particle size distribution. In some applications the set of fraction concentrations is used only as a raw data for the identification of the model spectrum (log-normal, modified gamma etc). This is not a rational way to process information. The true raw data is the set of directly measured signals which is used in the instrument to estimate the fraction concentrations. Some amount of original measurement information is contained in covariation matrix of measurement errors and it is lost in the standard multifraction presentation of the data. The better way of data processing is to identify the model spectrum directly on the basis of the set of measured signals. …… From history: (An AEL internal document written 1992, see http://ael.physic.ut.ee/tammet/AEL/AEL1992drafts.pdf)http://ael.physic.ut.ee/tammet/AEL/AEL1992drafts.pdf

29. Thank you for attention!