NASA Technical Reports Server (NTRS) 20050177237: A Posteriori Correction of Forecast and Observation Error Variances

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A POSTERIORI CORRECTION OF FORECAST AND OBSERVATION ERROR
VARIANCES
Leonid Rukhovets
SAIC and Global Modeling and Assimilation Office, NASA/Goddard Space Flight Center,
Greenbelt, MD, USA
1. Proposed method of total observation and Distributions of 0-F’s corresponding to
forecast error variance correction is based on rawinsonde heights and winds, aircraft
the assumption about normal distribution of winds, cloudtrack winds, surface heights and
“observed-minus-forecast” residuals (0-F), winds are close to the normal distribution.
where 0 is an observed value and F is usually The rawinsonde mixing ratio distribution
a short-term model forecast. This assumption can not be considered normal. The kurtosis
can be accepted for several types of value for mixing ratio observations is also
observations (except humidity) which are not very large.
grossly in error (Andersson and Jarvinen 2. If a random variable X has normal
1999, Dharssi et al. 1992, Hollingsworth et al. distribution, then according to the statistical
1986 , Jarvinen and Unden 1997, Lorenc rules probability of X to be within the
and Hammon 1988 ). interval (0, ao)i s:
Degree of nearness to normal distribution can P(0 < X <ao)= F(a) - F(0) = F(a) - 0.5 (1)
be estimated by the symmetry or skewness Here
(luck of symmetry) a3 = p3/03 X
and kurtosis a4 = do4- 3 F(x)= l/& le-‘’‘* dt ,
Here pj = i-order moment, <r is a standard -m
deviation. It is well known that for normal o - standard deviation of X, a - arbitrary
constant, F standard normal distribution
distribution a3 = Q = 0. -
function.
Table 1 contains a3 and Q for 0-F’s of
Suppose we have a number (percentage) of
several types of observations: rawinsonde
observations within some interval (0,ao).
heights, winds and mixing ratio, aircraft
If the number does not correspond to (l), it
winds, cloudtrack winds, and surface heights
(recast as upper air) and winds. Six-hour can mean we are using (J that is not standard
model forecasts (F) were obtained using the deviation for these observations .
Goddard Earth Observing System 4.0.3 However this number can be used to correct
assimilation run for October 1-31 ,2003. the standard deviations.
Figs. 1-7 show 0-F histograms for these
observations (without gross errors).
rawinhght rawinwind rawinhumd aircrftwind cldtrkwind srfheight srfwind
a3 0.13 - 0.01 -0.08 0.01 -0.57 0.02 0.04
a 0.87 -1.12 6.08 - 0.67 0.41 -1 -07 -1.74
Corresponding author address: Assimilation Oflice, Mail Code 610 .1 ,M D
Dr. Leonid Rukhovets, SAIC and 20771, USA
NASNGSFC, Global Modeling and
For this purpose it is convenient to apply so a result of the background check gave
called "background check" procedure which suspect observation percentage of 4.6. Then
is often used as a part of quality control in 1 - a/(2*100) = 0.977.
data assimilation systems (Anderson and From (3) and the standard normal
Jarvinen 1999, Dee et al. 1999) . distribution table we have 2*0"/0 = 2, and
The background check performs an a = 0". That is 0" is specified correctly.
examination of all observations against the But it corresponds to the known statistical
short range (6 hours) forecasts. Actually, the rule: about 4.6% of observations should be
inequality is verified for each observation: beyond 20.
(o-F)2 < a2 {(if0)2 + (0"f)2} (2) 4. Conclusions:
a) Using results of a background check
where a is a tolerance parameter, 0"" and
the prescribed statistics of (0"" )2 + (0"f)2
0" are respectively appropriate prescribed
can be corrected. Then the background
observation and forecast error standard
check can be repeated with the new
deviations. If for some observation the
a = +(af ) 2 } 1/2 .
inequality (2) is not fulfilled, the observation
b) The equation (4) together with the
is marked as suspect. The prescribed values
results of appropriate background check can
0"' and 0" can approximately represent
be considered as a relation between
the true observation and forecast error observation and forecast error standard
standard deviations. deviations. If the true value of a"i s known,
Let 0" = ((0"" >2 + and we can calculate of.
a = {(o'O)~ + (of)21/2} , where &a'nd of checc)k s fCoro tnhseid searm ree soubltsse rovfa ttiwoon tbyapcek, gbruotu fnodr
are true standard deviations. Instead of (2)
different instruments. Then using (4) we can
we can write:
write two equations for true sigmas :
I 0 - F I < a (0" / O)CJ (a$+(a;f 12 -- s,
Suppose M is the percentage of suspect
observations which was obtained by some +
(a,"), =s,
background check.
It means: Because of = a{ for both types of
P(0 < 0-F < ao")= 0.5 - M/(2*100) observations , subtraction of one equation
But according to (1) we have: from the other gives a relation between two
P(0 < 0-F < a s )= F(a0"/0) - 0.5 observation error statndard deviations. If one
Then of them can be found easier than the other,
F(ao"/(~=) 1 - M/(2*100) (3) the relation can be used to get the second
Using the table of standard normal one through the first one. For example,
distribution hnction we can find the value observation error standard deviation for
a o" /O = rn, corresponding to 1 - M/(2* 100). TOVS heights can be found through the
Then we can find o = ao"/rn (4) rawinsonde height observation error
3.Consider one example. Let a= 2. Suppose standard deviation. Analogously, cloud track
E
wind observation error standard deviation Hollingsworth, A. , D. B. Shaw,
can be found through the rawinsonde wind P.Lonnberg, L. Illari, K. Arpe and
observation error standard deviation. A.Simmons, 1986: Monitoring of
observation and analysis quality by '
REFERENCES : a data assimilation system. Moa
Anderson, E., and H. Jarvinen, 1999: Tea. Rev., 114, 861-879
Variational quality control. Q.1R . Jarvinen, H. , and P. Unden , 1997:
Meterol. Soc., 125, 697-722 Observation screening and first
Dharssi, A. C ., A. C. Lorenc, and guess quality control in the ECMWF
N. B. Ingleby, 1992: Treatment of 3D-Var data assimilation system.
gross errors using maximum ECMWF Tech. Memo.236.
probability theory. Q.J.R.Meteorl. Lorenc, A. C., and 0. Hammon, 1988:
SOC., 118,1017-1036. Objective quality control of
Dee, D.P., L. Rukhovets, R. Todling, A.M. observations using Bayesian
da Silva, and J.W.Larson, 1999: An methods. Theory and practical
adaptive buddy check for implementation.Q.J.R.Meteor1.S OC.
observational quality control. 11431 5-543
Q.J.R.Meteorl.Soc., 127,2451- 2471
x io' omf
-100 -50 0
Figure 1. Histogram of 0-F rawinsonde heights. Global. All levels. October 2003.
Number of observations = 457919. Yelow color corresponds to observations that were
marked as suspect by background check, but passed the adaptive buddy check (Dee et
all. 1999). The blue curve shows the Gaussian distribution for the same mean and
standard deviation.
OI ' ornf
6-
5-
4-
3-
2-
1-
0- A
-15 -10 -5 0 5 10 15
Figure 2. As in Fig.1 but fbr rawinsonde winds. Number of observations = 862146
omf
,
r , I
I I I I
10000 - I I I
I I I
9000 - I
8000 - I I I
- I I I
7000 I I
I I
I I I
6ooo- I I I I
I 1 I
I I I
5ooo- I I I
I I
4000 - I 1
I !
- I
3000 I
I
I
2ooo- I
I
1000 - I
I
" -25 -20 -15 -10 -5 0 5 10 15 20 2!
Figure 3. As in Fig. 1 but for aircraft winds. Number of observations = 148612
x IOi ornf
I I I
lean = -0.10362 I
tdv = 1.0053
- 5 4 - 3 - 2 - 1 0 1 2 3 4 5
Figure 4. As in Fig. 1 but for rawinsonde water vapor mixing ratio. Number of
observations = 198624.
ornf
1
5000
I ' I I I
I1 I I
4000 I I I I I
I I I I I
I I I I 1
I I I I I
t I 1 ' : I
3Mx) I I
I I I I
I I I
I
I
I
I
------?A d LA
-1 5 -10 -5 0 5 10 15 20
Figure 5. As in Fig.1 but for cloud track winds. Number of observations = 85776
I
ornf
L
-60 40 -20 0 20 40 60
Figure 6. As in Fig.1 but for surface geopotential heights. Number of observations =
463369.
omf
I I I I I I I I
I
I
I
I
I
I
I
I,
t
I
I
"i I
3ooo I
I
I
I
I
loo0 I I
I
I 5 -10 -5 0 5 10 15
Figure 7. As in Fig.1 but for surface winds. Number of observations = 123254.