Flux uncertainty and quality criteria · The random flux uncertainty depends in addition to...
Transcript of Flux uncertainty and quality criteria · The random flux uncertainty depends in addition to...
Flux uncertainty and qualitycriteria
Üllar Rannik, University of Helsinki
Outline
• Basic characteristics of turbulent records
• Flux random uncertainty
• Methods to calculate random errors
• Flux quality criteria
• Examples and statistics
Examples of turbulent records
Examples of probability distributions and correlation functions
An example of the “same flux“ measured by two different EC systems
• Measured at SMEAR II
• Two EC systems located approx. at 30 m distance
• Measuring almost the same flux footprint
• But not exactly the same realisations of turbulence
• Turbulence not (fully) independent
Random errors: general considerations• Imagine time series x with pdf p(x)
A) white noise (no correlation in time)
• Standard deviation (SD) of the average is the measure for the error due to finite ( ), random sampling:
• N is the number of independent observations
B) A time series with integral time scale
• In this case number of independent observations is decreased and
where the factor accounts for reduction of number of independent observations
In case of ensemble averaging no random error occurs!
2/1-= Nxx ssò¥
=0
2 ')'(1 dttRxx
x st
tNx
xx D=
tss
2
tx
Dt2
tTND
=
Random uncertainty of flux - definition
• If w(t), w(t+t), c(t) and c(t+t) are mutually joint Gaussian processes, then error variance of flux
><-= wcwcj
• Valid for
• Integral time scale
• Variance
• Covariance function
This is equivalent to
where
T<<jt
Flux uncertainty as random error (δ), being the measure of one standard deviation of the random uncertainty of turbulent flux observed over an averaging period T, can be evaluated by following different approaches:
Methods for estimating Flux uncertaintyMethods for estimating Flux uncertainty
( )[ ]22 ''''2
cwcwTIF -= jtd• “instantanous flux” (Wyngaard,
1973)
2/1-F= NSE sd• “standard error” (Vickers and
Mahrt, 1997)
ò¥
¥-
- += dffSfSfST wccwFM21 )()()(d• “Fourier method”(Rannik and
Vesala, 1999)
FrrorFlux RelativeNFF
RFE iFSEsd
==
Random errors, particle fluxes
The random flux uncertainty
depends in addition to turbulent flux variance also on the time-correlation of instantaneous flux events
Then the integral time scale of j is defined according to
(EQ. 1)
Where is the auto-covariance
function of φ.
))(('' ccwwcw --==j
ò¥
=0
2 ')'(1 dttRjj
j st
))'()()(()'( jjjjj -+-= ttttR
The integral time scale of fluxThe integral time scale of flux
( )[ ]22 ''''2
cwcwTIF -= jtd
A common parameterization used for this timescale is
where is the effective measurement height above the displacement height and u the mean wind speed (Pryor et al., 2007).
uze /=jt
jtWe estimated by numerical approach (EQ. 1) and we present
our results in term of normalized frequency
ez
uzn e
jpt2=
• n = 0.27+/-0.35 (0.008 as se) for unstable conditions
• for stable conditions÷÷ø
öççè
æ÷øö
çèæ+=
26.0
4.3121.0Lzn e
Using as an estimate of ,
implies that
uze / jt
16.021
==p
n
Here we have observed higher values for the corresponding frequency implying that times scale is smaller roughly by a factor of 2 in unstable (n = 0.27) and neutral conditions and up to 6 in very stable conditions.
However, for random uncertainty this implies the difference by square root of the factor.
( )[ ]22 ''''2
cwcwTIF -= jtd
uzn e
jpt2=
Another method to calculate random flux error
Flux uncertainty estimations: comparisonFlux uncertainty estimations: comparison
y = 0.965 x, R = 0.96 y = 0.974 x, R = 0.92
The best correlation was observed between the flux error estimates δSE and δIF .
In the following analysis the error estimate δIF is used.
EC system setup - SMEAR II
Measurement level: 23.3 m
CPC: TSI model 3010
Gas analyzer: LI-6262, LiCor Inc
Sonic anemometer: Solent 1012R2, Gill
Aerosol size distribution (from 3 to 500 nm particles in diameter) measurements were performed within the canopy (at 2 m height) using Differential Mobility Particle Sizer(DMPS) system.
EXAMPLE: LONGEXAMPLE: LONG--TERM AEROSOL PARTICLE TERM AEROSOL PARTICLE FLUX OBSERVATIONS AT SMEAR IIFLUX OBSERVATIONS AT SMEAR II
Formation of new aerosol particles
A day with no apparent particle formation but still significant variation in concentrations and fluxes in May 31 2002
About 32.5 % flux estimates emission
Frequency distribution of fluxes (measured by EC)
Flux uncertainty estimationsFlux uncertainty estimations
Typical random flux error (EC) in the order of 20%
Can be much larger depending on the instrumental noise (signal to noise ratio) and level of the variability due to “local” turbulent transfer and non-local forcings
For aerosol particle flux, the counting error due to discrete nature of aerosols is another source of random uncertainty [see e.g. Fairall, 1984]. The uncertainty of average flux can be evaluated as
TQcF wcounting sd =
, where Q is the volumetric flow rate through the counting device and T the averaging period (assuming that virtually all particles are counted). For a CPC with the flow rate Q = 1 LPM used in our study the typical flux error due to counting statistics is from 104 to 105 m-2 s-1, which is much smaller than the observed average flux values.
Additional random error source for Additional random error source for particle fluxesparticle fluxes
IFFDjt IFFD uZ /=jt SPFD ''cw ''cw ''cwFDFDFD IFFDjt IFFD uZ /=jt SPFD ''cw ''cw ''cwFDFDFDwithwith% % % withwith% % %
L < 0Unstable with estimated with
% passing %F< 0
100* vd/u*
% passing %F< 0
100* vd/u*
% passing %F< 0
100* vd/u*
RFE < 1 32.3 72.0 0.274 28.1 73.4 0.305 33.9 71.4 0.263
RFE< 0.5 17.6 82.0 0.596 13.3 85.5 0.821 20.1 80.6 0.494
RFE< 0.3 7.4 93.2 1.08 4.55 96.6 1.50 10.0 89.8 0.839
L > 0Stable % passing %
F < 0100* vd/u*
% passing %F < 0
100* vd/u*
% passing %F < 0
100* vd/u*
RFE < 1 38.1 76.0 0.247 30.1 78.9 0.300 40.3 74.8 0.240
RFE < 0.5 26.0 81.6 0.409 15.0 86.6 0.568 29.3 80.1 0.319
RE F< 0.3 15.6 87.5 0.556 4.8 92.6 0.937 19.2 85.6 0.444
IFFD jt IFFD Uze=jt SPFD
Particle flux data selection according to Particle flux data selection according to random flux error estimates and thresholdsrandom flux error estimates and thresholds
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Flux quality criteriaVariable Description Apply to Allowed
valuesReferences
Flux instationarity
Stationarity Co-variances FS < 0.30 Foken and Wichura, 1996
Flux intermittency
Intermittency Co-variances FI < 1 Mahrt et al., 1998
Friction velocity
turbulence well developed?
Co-variances U* > 0.1-0.3 Not discussed here
Kurtosis Is the probability distribution narrow
Single variable time series
1 < KU < 8 Vickers and Mahrt, 1997
Skewness Is the probability distribution skewed
Single variable time series
-2 < SK < 2 Vickers and Mahrt, 1997
σu/u*,
σT/T*,…
Integral turbulence characteristics
Single variable time series
ITC<0.3 Foken and Wichura, 1996
Spectra Power spectraCo-spectra
Visual inspection
Not discussed here
Quality statistics applied for time series and fluxes. X = W, C denotes time series of vertical wind speed and concentration with duration T = 30 min, Xi subrecord with duration T/N = 5 min (N = 6). Correspondingly X , 'X , Xs and ''CWF = are the time average and deviation from it,
standard deviation of X and flux for the entire record of duration T; ix , iXs and '' iii CWF = are
time average, standard deviation and flux calculated over i’th subrecord. denotes averaging
over N subrecord values and ( ) 2/122 ><-><= iiF FFi
s is the standard deviation of iF . XS
denotes the spectrum of X ( ò¥
¥-
= XX dffS s)( ) and WCS is the cross-spectrum.
Acronym Test statistic Definition Range of quality values
SK Skewness 33' -XX s (-1,1)
K Kurtosis 44' -XX s (2,5)
HM Haar mean [ ] [ ] 11 4/))min()(max(min||max -+ -- XXXX Xii s
< 2
HV Haar variance [ ]2221
max -
+- XXX ii
sss < 2
FI Flux instationarity 1)( -- FFFi < 1
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Flux (in)stationarity (Foken and Wichura, 1996)
• Measure the quality of co-variances
• Often about 40% of data omitted due to these, especially during night
– x = u, T, CO2, H2O, etc.
– The flux is often considered non-stationary if FS>0.3 and the Reynolds’ decomposition is not valid
5min 30min
30min
' ' ' '' '
w x w xFSw x
-=
The covariance calculated for the whole period (e.g.
30min)
The covariance calculated as a mean of the co-variances of
5min periods
Single time series quality criteria (1)
• Higher order moments (SK, K): possible instrument or recording problems and physical but unusual behaviour
• Haar transforms: discontinuities
Time series quality criteria (2): physical but unusual behavior
• A) Hard flagged by K and Haar variance
• B) Flagged by Haar mean and variance
• C) Flagged by Haar variance
Table 2. Percentages of observations not satisfying the time series quality criteria given in Table
1. Time series are denoted as following: W – vertical wind speed; P - particle concentration
measured by EC system; T – temperature; CO2- carbon dioxide concentration; H2O – water
vapour concentration. Statistics: SK –skewness; K – kurtosis; HM - Haar mean; HV – Haar
variance; Any – one or several of the statistics (SK, K, HM, HV) not satisfied. Total number of
analysed 30 min periods was 4933 for unstable (L < 0) and 5554 for stable (L > 0) stratifications.
Stability Quantity SK K HM HV Any L < 0
Unstable W 0.1 1.5 0.0 0.4 1.7 P 17.4 23.0 4.4 7.1 27.6 T 8.7 8.6 1.7 1.1 12.9
CO2 8.9 32.1 0.3 1.8 32.9 H2O 3.8 8.8 5.0 2.6 15.5
L > 0 Stable
W 0.4 5.7 0.0 1.3 5.9 P 12.2 18.5 2.0 3.8 21.0 T 3.2 5.5 0.7 0.9 6.9
CO2 15.4 46.7 0.5 1.6 47.5 H2O 4.3 19.5 2.0 1.0 21.7
Performance of (single) time series quality criteria
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Integral turbulence test
• Measure the quality of the time series of a single variable (Wichura and Foken, 1995)
• Is the turbulence well developed? Is the flux variance similarity followed?
• Normalized standard deviation for wind components and a scalar as a function of stability
2 2, ,
1 1* *
c cu v w xz d z dc cu L X L
s s- -æ ö æ ö= =ç ÷ ç ÷è ø è ø
An example from SMEAR IIIFrom Vesala et al. 2008a
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• If the measured normalized standard deviation deviates less than 30% from the model, the turbulence is considered well developed
mod*
*mod*
)/()/()/(
XXXITC
x
mesxx
sss
s-
=
From Lee et al. 2004, p.192Originally from other papers.
Parameter z/L c1 C2
σw/u* 0>z/L>-0.0032 1.3 0
-0.0032>z/L 2.0 1/8
σu/u* 0>z/L>-0.0032 2.7 0
-0.0032>z/L 4.15 1/8
σT/T* 0.02<z/L<1 1.4 -1/4
0.02>z/L>-0.062 0.5 -1/2
-0.062>z/L>-1 1.0 -1/4
-1>z/L 1.0 -1/3
• Instrumental noise – white noise
• Assumes no correlation btw. vertical wind speed and noise signal
• Could be estimated as (from std of noise c)
ncw
cw
sss =''
Performance of flux quality criteria
Table 3. Percentages of observations satisfying the scalar flux quality thresholds for different
statistics for unstable (L < 0) and stable (L > 0) stratification conditions. RFE – random flux
error; TRFE – theoretical random flux error; FI – flux intermittency. Number of all analysed 30
min periods 10400, unstable 4900, stable 5500.
Stability Scalar Threshold 1.0 Threshold 0.3
RFE TRFE FI RFE TRFE FI L < 0
Unstable P 76.2 57.6 72.0 33.6 8.2 30.4 T 91.4 88.2 91.8 71.3 64.4 73.9
CO2 94.7 93.3 98.3 79.7 69.4 78.7 H2O 94.4 88.0 92.5 72.7 49.2 67.5
L > 0 Stable
P 80.5 63.1 72.8 41.9 14.1 34.2 T 92.4 89.0 91.8 68.8 60.7 74.9
CO2 93.0 87.3 89.8 70.6 58.7 71.5 H2O 80.7 59.5 72.4 41.2 9.9 30.7
FI – flux (in)stationarity