KL-parameterization of atmospheric aerosol size distribution University of Tartu, Institute of Physics with participation of Marko Vana Acknowledgements to Markku Kulmala and staff of Hyytiälä station KL parameterization of atmospheric aerosol size distribution 1. Assimilation of information 2. KL-model of size distribution 3. Test data 4. Test results

A well forgotten model, references: Tammet, H.F. (1988) Sravnenie model'nykh raspredeleniï aérozol'nykh chastits po razmeram (in Russian). Acta Comm. Univ. Tartu 824, 92–108. Translation of the previous paper: Tammet, H. (1992) Comparison of model distributions of aerosol particle sizes. Acta Comm. Univ. Tartu 947, 136–149, Tammet, H. (1988) Models of size spectrum of tropospheric aerosol. In Atmospheric Aerosols and Nucleation. Lecture Notes in Physics, Springer-Verlag, Vienna, 309, pp. 75–78,

An example: two parameterizations

Assimilation of information Correlated parameters a ja b Spread ~S1 Lost information: Spread ~S2

Theory: 2D lost information: Equivalent error amplification: nD lost information: Equivalent error amplification: correlation matrix correlation coefficient

KL-model of size distribution Modification from 1988/92 to 2012:  radius replaced by diameter,  natural logarithm replaced by decimal logarithm.

3 variants of K

3 variants of L

3 variants of b

3 variants of d x

Analytic properties

Test data: origin and preparation Hyytiälä aerosol measurements downloaded by Marko Vana Three full years of 2008, 2009 ja files dmYYMMDD.sum 40-columns d = 3…983 nm 1051 files apsYYYYMMDD.sum 54-columns d = 523…19810 nm Time step 10 minutes, a file contains header and ja 144 lines of data. Some files contained broken lines or negative values of dn/dlgd, such files were rejected. Further, only these days were used where both DM and APS-files are present. Preparative operations:  DM & APS–files were joined using new logarithm-homogeneous fraction structure containing 62 fractions from 3 to nm (method – interpolation). Where both DM and APS data present the average was calculated using weights (d – 500) / 500 for APS and (1000 – d) / 500 for DMPS.  All diurnal files were merged into a single 3-year file while the time of an interval center was interpolated to sharp minute 5, 15, 25 … (using neighbors with deviation < 10.8 minutes).

Test data The file of 10-minute records Hyytiala08-10aerosol.xl contains 62 data columns and data lines that is 89.6 % of the maximum minute intervals in the 3 years. The 10-minute data are pretty noisy. Next, the data were convereted to hourly averages. Only these hours were included that contain at least 3 measurements. The file of hourly averages Hyytiala08-10aerosol-h.xl contains one header line (incl. diameters) and data lines that cover 89.4% of possible hours. The 71 columns are: time DOY, 62 values of dn/dlgd, total number concentration, time parameters: year, month, day, hour, year quarter, day quarter, day-of-week.

Test data

Some strange irregularities:  the interval 10…..165 nm contains dn/dlgd < 10 cm -3,  the interval 190…3000 nm contains dn/dlgd < cm -3. Such hours were excluded from the final KL test data. The file of filtered hourly averages Hyytiala08-10aerosol-hf.xl (h = hours, f = filtered) contains hours that is 82.5% of possible maximum. Filtered test data

KL-parameterization of atmospheric aerosol size distribution TEST RESULTS

3-year average: KL4 & KL5 from 10 to 3000 nm

What is KL5?

3-year average: KL4 & KL5 from 3 to nm KL4: K = 3.18 L = 0.97 b = 4240 dx = 117 std = KL5: K = 3.05 L = 1.01 b = 4980 dx = 98 c = 0.45 std = 0.138

Method of fitting Given: a table of function measured _dn/dlgd (d) Task: choose 5 parameters K L b d x c Special case of KL4  c = 0. The fitting deviation in typical diagrams is Δ = lg (fitted_dn/dlgd) – lg (measured_dn/dlgd). Measure of visual quality: std (Δ) Policy:  choose b so that average (Δ) = 0  choose other parameters so that std (Δ)  min. An arbitrary technique of minimization can be used

Fitting of test data Mean standard deviation between approximation and measurements of lg (dn/dlgd) KL4  KL5  Standard deviation of mean distribution approximation were: KL4  KL5  0.032

Examples: 10% of KL4 9500) KL4: KL5:

Examples: 10% of KL ) KL KL

Examples: 10% of KL ) KL KL

Examples: 50% of KL ) KL KL

Examples: 50% of KL ) KL KL

Examples: 50% of KL ) KL KL

Examples: 90% of KL ) KL KL

Examples: 90% of KL ) KL KL

Examples: 90% of KL ) KL KL

Analysis: KL4 Correlation matrix K L b d x Det , loss 0.36 digits, error amplification 1.23 Eigenvectors Eigenvalues

Analysis: KL5 Correlation matrix K L b d x c Det , loss 0.39 digits, error amplification 1.25 Eigenvectors Eigenvalues

Conclusion: it works outx  graphic,  simple interpretation,  minimum loss of information,  analytic integrals available. Properties of KL:

2009 THANK YOU, KL