KL parameterization
of atmospheric aerosol
size distribution
KL-parameterization of atmospheric aerosol size distribution
1. Assimilation of information
2. KL-model of size distribution
3. Test data
4. Test results
Hannes.Tammet@ut.ee
University of Tartu, Institute of Physics
with participation of Marko Vana
Acknowledgements to Markku Kulmala and staff of Hyytiälä station
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,
http://ael.physic.ut.ee/tammet/kl1992.pdf
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,
http://www.springerlink.com/content/p7l948j8m356605k
2000
f (d )
dN
f (d ) 
dd
1000
0
0
500
d
1000
An example: two parameterizations
a   f (d )dd
b   d  f (d )dd
a   f (d )dd
b
d

f
(
d
)
dd


f (d )dd
Assimilation of information
Correlated parameters a ja b
Lost information:
S2
I  log
S1
Spread
~S2
Spread
~S1
Theory:
2D
lost information:
correlation
coefficient
S2 1
1
I  log
 log
S1 2
1 R2
Equivalent error amplification:
nD
lost information:
S2 4 1
K

2
S1
1 R
V2 1
1
I  log
 log
V1 2
det C
1 (2n)
Equivalent error amplification:
 1 
K 

 det C 
correlation
matrix
KL-model of size distribution
Modification from 1988/92 to 2012:
 radius replaced by diameter,
 natural logarithm replaced by decimal logarithm.
dN
dN
b
f lg d (d ) 
 ln 10

K
d lg d
d ln d  d   d  L
     
 d   d 
3 variants of
K
10000
M
1000
KKL
100
dn/dlogd
K+
10
1
0.1
0.01
0.001
10
100
1000
d : nm
10000
3 variants of
L
10000
M
1000
KL
L-
100
dn/dlogd
L+
10
1
0.1
0.01
0.001
10
100
1000
d : nm
10000
3 variants of
b
10000
M
1000
KL
b-
100
dn/dlogd
b+
10
1
0.1
0.01
0.001
10
100
1000
d : nm
10000
3 variants of
dx
10000
M
1000
KL
dx-
100
dn/dlogd
dx+
10
1
0.1
0.01
0.001
10
100
1000
d : nm
10000
Analytic properties
f lg d (d ) 
d ave
L
sin
KL d

(1  L) 
sin
KL
d
L
ˆ
d  
K
M q   d f d (d )dd 
q
N  M0 
d 
K
b
ln 10  ( K  L) sin
L
KL
1
K L
b
L
 d d 
L
K L
d
L
ˆ
f lg d (d )  b
K
K L
K
KL
b
ln 10  ( K  L) sin
L
KL

2bd3
V  M3 
(3  L)
6
6 ln 10  ( K  L) sin
KL
Test data:
origin and preparation
Hyytiälä aerosol measurements downloaded by Marko Vana
Three full years of 2008, 2009 ja 2010
1107 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 19110 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 141367 data lines that is 89.6 % of the maximum
157824 10-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 23517 data lines
that cover 89.4% of possible 26304 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
10000
1000
dn/dlogd
100
10
1
f(d)
f(d)d/100
0.1
f(d)d2/20000
f(d)d3/5000000
0.01
0.001
1
10
100
1000
d : nm
10000
Test data
2000
f(d)
f(d)d/100
f(d)d2/20000
1500
dn/dlogd
f(d)d3/5000000
1000
500
0
1
10
100
1000
d : nm
10000
Test data
10000
ave
std
1000
std/ave
dn/dlogd
100
10
1
0.1
0.01
0.001
1
10
100
1000
10000 d : nm
100000
Filtered test data
Some strange irregularities:
 the interval 10…..165 nm contains dn/dlgd < 10 cm-3,
 the interval 190…3000 nm contains dn/dlgd < 0.001 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 21682 hours that is 82.5% of possible maximum.
KL-parameterization of atmospheric aerosol size distribution
TEST
RESULTS
3-year average: KL4 & KL5 from 10 to 3000 nm
10000
Measurements
1000
KL4
dn/dlogd
100
KL5
10
1
KL4:
K = 3.12 L = 0.45 b = 2550 dx = 138 std = 0.097
0.1
KL5:
K = 2.78 L = 0.77 b = 3750 dx = 94 c = 0.59 std = 0.032
0.01
0.001
1
10
100
1000
d : nm
10000
What is KL5?
dn/dlgd / fKL4d
2.5
ave
median
2
std
approx
1.5
1
d
0.5
10
100
1000
10000


d 
d 
2
2
f c  3 exp   3 ln 
   exp   0.5 ln 
 
 250nm  
 250nm  


b1  cf c (d ) 
dN
f lg d (d ) 

d lg d d d K  d d L
3-year average: KL4 & KL5 from 3 to 10000 nm
10000
ave
1000
KL4
100
dn/dlogd
KL5
10
1
0.1
KL4:
K = 3.18 L = 0.97 b = 4240 dx = 117 std = 0.152
KL5:
K = 3.05 L = 1.01 b = 4980 dx = 98 c = 0.45 std = 0.138
0.01
0.001
1
10
100
1000
d : nm
10000
Method of fitting
Given: a table of function measured _dn/dlgd (d)
Task: choose 5 parameters K L b dx 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
15
%
KL4
10
KL5
5
std
0
0
0.04
0.08
0.12
0.16
0.2
0.24
0.28
0.32
0.36
Mean standard deviation between approximation and
measurements of lg (dn/dlgd)
KL4  0.192
KL5  0.144
Standard deviation of mean distribution approximation were:
KL4  0.097
KL5  0.032
Examples: 10% of KL4
9500) 2009-04-14-12
KL4: 0.109 3.681 0.265 1149 267
KL5: 0.106 3.616 0.229 1073 266 0.119
10000
Measured
1000
KL4
KL5
dn/dlogd
100
10
1
0.1
0.01
0.001
10
100
1000
d : nm
10000
Examples: 10% of KL4
12752) 2009-09-30-05
KL4 0.109 2.006 2.226 6407 18
KL5 0.094 2.000 2.446 6557 18 0.236
10000
Measured
1000
KL4
KL5
dn/dlogd
100
10
1
0.1
0.01
0.001
10
100
1000
d : nm
10000
Examples: 10% of KL4
11805) 2009-08-19-07
KL4 0.109 3.026 2.051 7109 70
KL5 0.108 2.924 2.33 7878 62 0.443
10000
Measured
1000
KL4
KL5
dn/dlogd
100
10
1
0.1
0.01
0.001
10
100
1000
d : nm
10000
Examples: 50% of KL4
11714) 2009-08-11-10
KL4 0.189 3.16 1.785 7544 114
KL5 0.095 2.795 2.353 9634 81 0.973
10000
Measured
1000
KL4
KL5
dn/dlogd
100
10
1
0.1
0.01
0.001
10
100
1000
d : nm
10000
Examples: 50% of KL4
11844) 2009-08-20-23
KL4 0.189 3.05 2.383 11169 55
KL5 0.189 3.044 2.403 11216 55 0.033
10000
Measured
1000
KL4
KL5
dn/dlogd
100
10
1
0.1
0.01
0.001
10
100
1000
d : nm
10000
Examples: 50% of KL4
11915) 2009-08-24-04
KL4 0.189 3.102 2.526 4226 95
KL5 0.102 2.855 2.982 4724 76 0.879
10000
Measured
1000
KL4
KL5
dn/dlogd
100
10
1
0.1
0.01
0.001
10
100
1000
d : nm
10000
Examples: 90% of KL4
18557) 2010-08-11-13
KL4 0.278 3.19 0.361 2101 163
KL5 0.172 2.663 0.745 3737 96 1.312
10000
Measured
1000
KL4
KL5
dn/dlogd
100
10
1
0.1
0.01
0.001
10
100
1000
d : nm
10000
Examples: 90% of KL4
18549) 2010-08-11-05
KL4 0.222 3.235 1.197 1472 154
KL5 0.182 2.615 1.954 2418 85 1.335
10000
Measured
1000
KL4
KL5
dn/dlogd
100
10
1
0.1
0.01
0.001
10
100
1000
d : nm
10000
Examples: 90% of KL4
19310) 2010-09-14-12
KL4 0.278 2.895 1.179 1293 104
KL5 0.120 2.523 2.127 2389 62 1.409
10000
Measured
1000
KL4
KL5
dn/dlogd
100
10
1
0.1
0.01
0.001
10
100
1000
d : nm
10000
Analysis: KL4
Correlation matrix
K
L
b
1.000 -0.251 -0.207
-0.251 1.000 0.594
-0.207 0.594 1.000
0.717 -0.567 -0.446
dx
0.717
-0.567
-0.446
1.000
Det 0.191055, loss 0.36 digits, error amplification 1.23
0.449
-0.504
-0.460
0.575
Eigenvectors
0.666 -0.198
0.427 -0.668
0.539 0.704
0.286 0.132
-0.561
0.340
0.022
0.754
2.413
Eigenvalues
0.978 0.410
0.197
Analysis: KL5
K
1.000
-0.286
-0.152
0.730
-0.393
Correlation matrix
L
b
dx
-0.286 -0.152 0.730
1.000 0.589 -0.546
0.589 1.000 -0.373
-0.546 -0.373 1.000
0.215 0.071 -0.336
c
-0.393
0.215
0.071
-0.336
1.000
Det 0.167549, loss 0.39 digits, error amplification 1.25
0.470
-0.473
-0.377
0.553
-0.324
Eigenvectors
0.432 -0.422 -0.220
0.418 -0.154 -0.699
0.619 -0.161 0.669
0.125 -0.354 0.090
-0.487 -0.803 0.076
0.602
-0.295
-0.017
-0.737
0.068
2.540
Eigenvalues
1.159 0.701 0.392
0.207
Properties of KL:
 graphic,
 simple interpretation,
 minimum loss of information,
 analytic integrals available.
Conclusion:
it works outx
2009
THANK
YOU,
KL